Radii of starlikeness and convexity of generalized Struve functions
Evrim Toklu





E. Toklu]Evrİm Toklu
Department of Mathematics, Faculty of Education, Ağrı İbrahım Çeçen University, 04100 Ağrı, Turkey 
evrimtoklu@gmail.com



 
 arabic@̧footnote

Radii of starlikeness and convexity of generalized Struve functions 
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In this paper, it is aimed to determine the radii of starlikeness and convexity of the normalized generalized Struve functions for three different kinds of normalization and to find tight lower and upper bounds for the radius of starlikeness and convexity of these normalized Struve functions by making use of Euler-Rayleigh inequalities. The Laguerre-Pólya class of entire functions has a crucial role in constructing our main results.




§ INTRODUCTION AND PREREQUISITES

As it is well known, special functions are one of the most powerful tools in the solution of a wide variety of important problems. Because of the fruitful properties of special functions, it is important to examine their properties in many aspects. In the recent years, there has been a vivid interest on geometric properties of special functions from the point of view of geometric function theory, like Bessel, Struve and Lommel functions of the first kind; see the papers <cit.> and the references therein. However, it is possible to say that the roots of special functions seen on geometric functions theory are based on the studies of Brown <cit.>, Kreyszig and Todd <cit.> and Wilf <cit.> which initiated investigation on the univalence of Bessel functions and on the determination the radius of starlikeness for different kinds of normalization. Recently, in 2014, Baricz et al. <cit.> came up with a much simpler approach for determining the radius of starlikeness of normalized Bessel functions of first kind. In the same year, Baricz and Szász <cit.> obtained the radius of convexity of the same normalized Bessel functions. When scrutinized their studies (in particular,  <cit.> and <cit.>) the main facts pertaining to these studies can be stated as follows: the radii of univalence, starlikeness and convexity are obtained as solutions of some transcendental equations and the obtained radii satisfy some interesting inequalities. In addition, the other main fact is that the positive zeros of Bessel, Struve, Lommel functions of the first kind and the Laguerre-Pólya class of entire functions have an important place in these papers. Recent years, there has been extensive work on determining some geometric properties of other special functions involving Bessel function of first kind such as univalence, starlikeness, convexity and so forth. For instance, in <cit.>, Deniz and Szász obtained the radius of uniform convexity of the normalized Bessel functions. And also, Bohra and Ravichandran in <cit.> investigated the radius of strong starlikeness and k uniform convexity of order α of the normalized Bessel functions.

Motivated by the above series of papers on geometric properties of special functions and using the method of Baricz et al. <cit.>, our aim in this paper is to present some similar results for the normalized forms of the generalized Struve functions, which is similar to the generalized Bessel functions of Wright type (and of Galue type).

This paper is organized as follows: The rest of this section contains some basic definitions needed for the proof of our main results. Section 2 is divided into three subsections: The first subsection is devoted to the study on the radii of starlikeness of normalized generalized Struve functions. The second subsection contains the study of the radii of convexity of  normalized generalized Struve functions. The last subsection is allocated to the presentation of making some comparisons our main results given in this paper with obtained results earlier. 

Before starting to present our main results, we would like to draw attention to some basic concepts needed for building our main results. For r>0 we denote by 𝔻_r={z∈ℂ: |z|<r} the open disk of radius r
centered at the origin. Let f:𝔻_r→ℂ be the function defined by

    f(z)=z+∑_n≥ 2a_nz^n,

where r is less or equal than the radius of convergence of the above power series. Denote by 𝒜 the class of allanalytic functions of the form (<ref>), that is, normalized by the conditions f(0)=f^'(0)-1=0. We say that the function f, defined by (<ref>), is starlike function in 𝔻_r if f is univalent in 𝔻_r, and the image domain f(𝔻_r) is a starlike domain in ℂ with respect to the origin (see <cit.> for more details). Analytically, the function f is starlike in 𝔻_r if and only if 
    ( zf^'(z)/f(z)) >0     z∈𝔻_r.
 For α∈ [0,1) we say that the function f is starlike of order α in 𝔻_r if and only if 

    ( zf^'(z)/f(z)) >α    z∈𝔻_r.

The radius of starlikeness of order α of the function f is defined as the real number

    r_α^⋆(f)=sup{ r>0 |( zf^'(z)/f(z))  >α for all z∈𝔻_r.}.

Note that r^⋆(f)=r_0^⋆(f) is in fact the largest radius such that the image region f(𝔻_r^⋆(f)) is a starlike domain with respect to the origin.

The function f, defined by (<ref>), is convex in the disk 𝔻_r if f is univalent in 𝔻_r, and the image domain f(𝔻_r) is a convex domain in ℂ. Analytically, the function f is convex in 𝔻_r if and only if

    (  1+zf^''(z)/f^'(z))>0      z∈𝔻_r.

For α∈[0,1) we say that the function f is convex of order α in 𝔻_r if and only if 

    ( 1+zf^''(z)/f^'(z))
    >α    z∈𝔻_r.
 
We shall denote the radius of convexity of order α of the function f by the real number

    r_α^c(f)=sup{ r>0 |( 1+
    zf^''(z)/f^'(z)) >α for all 
    z∈𝔻_r.} .

Note that r^c(f)=r_0^c(f) is the largest radius such that the image region f(𝔻_r^c(f)) is a convex domain.

We recall that a real entire function q belongs to the  Laguerre-Pólya class ℒ𝒫 if it can be represented in the form 
    q(x)=cx^me^-ax^2+bx∏_n≥1(1+x/x_n)e^-x/x_n,
 with c,b,x_n∈ℝ, a≥0, m∈ℕ_0 and ∑1/x_n^2<∞. We note that the class ℒ𝒫 is the complement of the space of polynomials whose zeros are all real in the topology induced by the uniform convergence on the compact sets of the complex plane of polynomials with only real zeros. For more details on the class ℒ𝒫 we refer to <cit.> and to the references therein.


Finally, let us take a look at some lemmas which are very useful in building our main results.


If  a>b>r≥|z|, and λ∈ [0,1], then

    |z/b-z-λz/a-z|≤r/b-r-λr/a-r.

The followings can be obtained as a natural consequence of this inequality:

    (z/b-z-λz/a-z)≤r/b-r-λr/a-r

and 

    (z/b-z)≤|z/b-z|≤r/b-r.



The result of Runckel <cit.>, stated in <Ref>, play a key role in proving the reality of zeros of some special functions. For interesting applications of this lemma one can refer to the study of Baricz and Sanjeev <cit.>.

If f(z)=∑_n≥ 0a_nz^n can be represented by f(z)=e^az^2h(z), where a≤0 and h is of form

    h(z)=ce^bz∏_n≥ 1( 1-z/c_n)e^z/c_n,   a,b∈ℝ, ∑_n ≥ 1| c_n |^-2<∞,

f has real zeros only (or no zeros at all), and G is of form

    G(z)=e^β z∏_n≥1(1+z/α_n)e^-z/α_n,   α_n>0, β∈ℝ, ∑_n ≥ 1α_n^-2<∞,

then the function ∑_n≥0a_nG(n)z^n has real zeros only.




§ THE RADII OF STARLIKENESS AND CONVEXITY OF GENERALIZED STRUVE FUNCTIONS


In this section we focus on the generalized Struve functions, which is similar to the generalized Bessel functions of Wright type (and of Galue type), defined as

    _qW_p,b,c,δ(z)=∑_n≥ 0(-1)^nc^n/n!Γ(qn+p/δ+b+2/2)( z/2) ^2n+p+1,    (q∈ℕ;   p,b,c∈ℂ)

where δ>0.

It is clear that we can derive a number of well-known special functions from  generalized Struve function for some values of the parameters. Some of them is as follows:

    _1W_ν-1,2,1,1(z)=J_ν(z),

where J_ν(z) is Bessel function of first kind <cit.>.

    _qW_2ν+2λ-1,1,1,2(z)=( z/2)^νΓ(λ+n+1)/Γ(1+n)J_ν,λ^q(z),

where J_ν,λ^q(z) is Bessel-maitland function <cit.>.

    _qW_p-1,1,-1,1(z)=_qI_p(z),

where _qI_p(z) is the Galue type generalization of modified Bessel function <cit.>.

It is obvious that the function z↦_qW_p,b,c,δ(z) does not belong to the class 𝒜, and thus first we perform some natural normalization. We define three functions originating from _qW_p,b,c,δ(.)


    _qf_p,b,c,δ(z)=   (2^p+1Γ(p/δ+b+2/2)_qW_p,b,c,δ(z))^1/p+1, 
    _qg_p,b,c,δ(z)=   2^p+1Γ(p/δ+b+2/2)z^-p_qW_p,b,c,δ(z),
    _qh_p,b,c,δ(z)=    2^p+1Γ(p/δ+b+2/2)z^1-p+1/2_qW_p,b,c,δ(√(z)).

Observe that these functions belong to the class  𝒜. No doubt it is possible to write infinitely many other normalizations; the main motivation to consider the above ones is the fact that their particular cases in terms of Bessel functions appear in the literature or are similar to the studied normalization in the literature. In this study, we are going to present some intriguing results on these three functions by using some basic techniques of geometric function theory.


The following lemma, which we believe is of independent interest, has a crucial role in the proof of our main results.
 
Let δ,q,b,c>0 and p+1>0. Then z ↦_qW_p,b,c,δ(z) possesses infinitely many zeros which are all real. Denoting by _qω_p,b,c,n the nth positive zero of _qW_p,b,c,δ(z), under the same conditions the Weierstrassian decomposition

    2^p+1Γ(p/δ+b+2/2)_qW_p,b,c,δ(z)=z^p+1∏_n≥ 1(1-z^2/_qω_p,b,c,δ,n^2)

is fulfilled, and this product is uniformly convergent on compact subsets of the complex
plane. Moreover, if we denote by _qω_p,b,c,δ,n^' the nth positive zero of _qW_p,b,c,δ^'(z), then positive zeros of _qW_p,b,c,δ are interlaced with those of _qW_p,b,c,δ^'. In other words, the zeros satisfy the chain of inequalities

    _qω_p,b,c,δ,1^'<_qω_p,b,c,δ,1<_qω_p,b,c,δ,2^'<_qω_p,b,c,δ,2<…_.



Let us prove the reality of zeros of the function _qW_p,b,c,δ(z). Consider the entire function 

    _qW_p,b,c,δ(z)=(z/2)^p+1∑_n≥0(-1)^n c^n/n!Γ(qn+p/δ+b+2/2)(z/2)^2n .

The function _qG_p,b,c,δ:[0, ∞) →ℝ defined by

    _qG_p,b,c,δ(z)=1/Γ(qz+p/δ+b+2/2)

is entire function and of growth order 1, belongs to ℒ𝒫. If we choose f(z)=e^-c(z2) ^2, then by making use of the Runckel's above-mentioned result (that is, <Ref>) we obtain that the function _qW_p,b,c,δ(z) has real zeros only if δ,q,b,c>0 and p+1>0. Moreover, the growth order of the generalized Struve function is calculated in the following:

    ρ   =lim sup_n→∞n log n/-log|c_n | =lim sup_n→∞n log n/-log|(-c)^n/n!Γ(qn+p/δ+b+2/2)|
       =lim sup_n→∞1/-n log| c| /n log n+logΓ(n+1) /n log n+logΓ(qn+p/δ+b+2/2)/n log n
       =1/1+q.

It is evident that 0<ρ=1/1+q<1, for q>0. It is well known that if the finite growth order ρ of an entire function is not an integer, then the function has infinitely many zeros. In that case, by virtue of the Hadamard theorem on growth order of the entire function, it follows that its infinite product representation is exactly what we have in <Ref>. Taking into account the infinite product representation we get

    _qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z)=p+1/z-∑_n ≥ 12z/_qω_p,b,c,δ,n^2 -z^2.

Diferentiating both sides of (<ref>) we arrive at

    d/dz(_qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z))=-p+1/z^2-2∑_n ≥ 1_qω_p,b,c,δ,n^2+z^2/( _qω_p,b,c,δ,n^2-z^2) ^2.

Since the expression on the right-hand side is real and negative for z real, the quotient _qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z) is a strictly decreasing function from +∞ to -∞ as z increases through real values over the open interval (_qω_p,b,c,δ,n, _qω_p,b,c,δ,n+1)    n∈ℕ. That is to say that the function _qW_p,b,c,δ'(z) vanishes just once between two consecutive zeros of the function _qW_p,b,c,δ(z).

The lemma is proved.



 §.§ The radii of starlikeness of order α of the functions _qf_p,b,c,δ, _qg_p,b,c,δ and _qh_p,b,c,δ.

Our first main result is related to the radii of starlikeness of these three normalized generalized Struve functions. In other words, the aim of this section is to determine the radii of starlikeness of order α of the normalized Struve functions and to give tight lower and upper bounds for radii of starlikeness of the normalized Struve functions. 



Let δ,q,b,c>0, p+1>0 and α∈[ 0,1). Then the following assertions are true.

  a. The radius of starlikeness of order α of the function _qf_p,b,c,δ is r^⋆_α(_qf_p,b,c,δ)=_qx_p,b,c,δ,1, where _qx_p,b,c,δ,1 is the smallest zero of the equation

    r _qW_p,b,c,δ'(r)-α(p+1)_qW_p,b,c,δ(r)=0  .


  b. The radius of starlikeness of order α of the function _qg_p,b,c,δ is r^⋆_α(_qg_p,b,c,δ)=_qy_p,b,c,δ,1, where _qy_p,b,c,δ,1 is the smallest zero of the equation

    r _qW_p,b,c,δ'(r)-(α+p)_qW_p,b,c,δ(r)=0 .


  c. The radius of starlikeness of order α of the function _qh_p,b,c,δ is r^⋆_α(_qh_p,b,c,δ)=_qz_p,b,c,δ,1, where _qz_p,b,c,δ,1 is the smallest zero of the equation

    √(r)_qW_p,b,c,δ'(√(r))-2(α+p+1/2-1)_qW_p,b,c,δ(√(r))=0.



We shall show that the inequalities

    ( z_qf_p,b,c,δ'(z)/_qf_p,b,c,δ(z))>α,   (z_qg_p,b,c,δ'(z)/_qg_p,b,c,δ(z))>α and  ( z_qh_p,b,c,δ'(z)/_qh_p,b,c,δ(z))>α,

are valid for z∈𝔻__qx_p,b,c,δ,1(_qf_p,b,c,δ),  z∈𝔻__qy_p,b,c,δ,1(_qg_p,b,c,δ) and  z∈𝔻__qz_p,b,c,δ,1(_qh_p,b,c,δ), respectively, and
each of the above-mentioned inequalities does not hold in any larger disk. Consider the functions

    _qf_p,b,c,δ(z)=   (2^p+1Γ(p/δ+b+2/2)_qW_p,b,c,δ(z))^1/p+1, 
    _qg_p,b,c,δ(z)=   2^p+1Γ(p/δ+b+2/2)z^-p_qW_p,b,c,δ(z),
    _qh_p,b,c,δ(z)=    2^p+1Γ(p/δ+b+2/2)z^1-p+1/2_qW_p,b,c,δ(√(z)).

As a result of the logarithmic derivation we arrive at

    z_qf_p,b,c,δ'(z)/_qf_p,b,c,δ(z)=   1/p+1(z_qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z))=1-1/p+1∑_n ≥ 12z^2/_qω_p,b,c,δ,n^2-z^2,  
    z_qg_p,b,c,δ'(z)/_qg_p,b,c,δ(z)=   -p+(z_qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z))= 1-∑_n ≥ 12z^2/_qω_p,b,c,δ,n^2-z^2, 
    z_qh_p,b,c,δ'(z)/_qh_p,b,c,δ(z)=   1-p+1/2+1/2(√(z)_qW_p,b,c,δ'(√(z))/_qW_p,b,c,δ(√(z)))=1-∑_n ≥ 1z/_qω_p,b,c,δ,n^2-z.

By making use of Eq. (<ref>) given in <Ref> we have

    (z_qf_p,b,c,δ'(z)/_qf_p,b,c,δ(z))   =1-1/p+1(∑_n ≥ 12z^2/_qω_p,b,c,δ,n^2-z^2)  
    
       ≥ 1-1/p+1∑_n ≥ 12| z| ^2/_qω_p,b,c,δ,n^2-| z| ^2 = | z| _qf_p,b,c,δ'(| z| )/_qf_p,b,c,δ(| z| ),
    (z_qg_p,b,c,δ'(z)/_qg_p,b,c,δ(z))   =1-(∑_n ≥ 12z^2/_qω_p,b,c,δ,n^2-z^2) 
    
       ≥ 1- ∑_n ≥ 12| z| ^2/_qω_p,b,c,δ,n^2-| z| ^2= | z| _qg_p,b,c,δ'(| z| )/_qg_p,b,c,δ(| z|),
    
    ( z_qh_p,b,c,δ'(z)/_qh_p,b,c,δ(z))   =1-( ∑_n ≥ 1z/_qω_p,b,c,δ,n^2-z)
       ≥ 1- ∑_n ≥ 1| z| /_qω_p,b,c,δ,n^2-| z| =| z| _qh_p,b,c,δ'(| z| )/_qh_p,b,c,δ(| z| ).

It is important to mention that equalities in the above-mentioned inequalities are attained only when z=| z|=r. In light of the later inequalities and the minimum principle for harmonic functions we deduce that the inequalities stated in (<ref>) hold if and only if |z|<_qx_p,b,c,δ,1, |z|<_qy_p,b,c,δ,1 and |z|<_qz_p,b,c,δ,1, respectively, where _qx_p,b,c,δ,1, _qy_p,b,c,δ,1 and _qz_p,b,c,δ,1 are the smallest positive roots of the equations

    r_qf_p,b,c,δ'(r)/_qf_p,b,c,δ(r)=α,   r_qg_p,b,c,δ'(r)/_qg_p,b,c,δ(r)=α and  r_qh_p,b,c,δ'(r)/_qh_p,b,c,δ(r)=α.


As a natural result of these equalities we conclude that 

    r _qW_p,b,c,δ'(r)-α(p+1)_qW_p,b,c,δ(r)=0,    r _qW_p,b,c,δ'(r)-(α+p)_qW_p,b,c,δ(r)=0

and

    √(r)_qW_p,b,c,δ'(√(r))-2(α+p+1/2-1)_qW_p,b,c,δ(√(r))=0.

In order to finish the proof we need to show that the roots _qx_p,b,c,δ,1, _qy_p,b,c,δ,1 and _qz_p,b,c,δ,1 are, respectively, the smallest zeros of the above-mentioned transcendental equation. To do this, in view of the above inequalities, consider the functions for r∈( 0,_qw_p,b,c,δ,1)

    inf_z∈𝔻_r(z_qf_p,b,c,δ'(z)/_qf_p,b,c,δ(z))=   r_qf_p,b,c,δ'(r)/_qf_p,b,c,δ(r)=_qu_p,b,c,δ(r),
    inf_z∈𝔻_r( z_qg_p,b,c,δ'(z)/_qg_p,b,c,δ(z)) =   r_qg_p,b,c,δ'(r)/_qg_p,b,c,δ(r)=_qv_p,b,c,δ(r), 
    inf_z∈𝔻_r( z_qh_p,b,c,δ'(z)/_qh_p,b,c,δ(z)) =   r_qh_p,b,c,δ'(r)/_qh_p,b,c,δ(r)=_qw_p,b,c,δ(r).

Taking into account facts that

    _qu_p,b,c,δ'(r)=-1/p+1∑_n ≥ 14z_qω_p,b,c,δ,n^2/(_qw_p,b,c,δ,n^2-z^2 )^2 <0,   _qv_p,b,c,δ'(r)=-∑_n ≥ 14z_qω_p,b,c,δ,n^2/(_qω_p,b,c,δ,n^2-z^2 )^2 <0

and

    _qw_p,b,c,δ(r)=-∑_n ≥ 1_qω_p,b,c,δ,n^2/(_qω_p,b,c,δ,n^2-z )^2 <0

we deduce that the real functions _qu_p,b,c,δ, _qv_p,b,c,δ, _qw_p,b,c,δ:(0,_qω_p,b,c,δ,1)→ℝ are strictly decreasing. In addition, by virtue of the limits determined as

    lim_r ↘ 0_qu_p,b,c,δ(r)=lim_r ↘ 0_qv_p,b,c,δ(r)=lim_r ↘ 0_qw_p,b,c,δ(r)=1

and

    lim_r↗_qω_p,b,c,δ,1_qu_p,b,c,δ(r)=lim_r↗_qω_p,b,c,δ,1_qv_p,b,c,δ(r)=lim_r↗_qω_p,b,c,δ,1_qw_p,b,c,δ(r)=-∞.

we say that the roots _qx_p,b,c,δ,1, _qy_p,b,c,δ,1 and _qz_p,b,c,δ,1 are, respectively, the smallest zeros of the transcendental equation given in (<ref>). It is clear that these are desired results.


The following theorems include some tight lower and upper bounds for the radii
of starlikeness of the functions considered in the above theorems.

Let δ,q,b,c>0 and p+1>0. The radius of starlikeness r^⋆(_qf_p,b,c,δ) is satisfies

    √(2(p+1)Γ(q+p/δ+b+2/2)/c(p+3)Γ(p/δ+b+2/2))<r^⋆(_qf_p,b,c,δ)<2√((p+1)(p+3)Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{(p+3)^2Γ(p/δ+b+2/2)Γ(2qp/δ+b+2/2)-2(p+5)(p+1)Γ^2(q+p/δ+b+2/2)}).



When α=0 in <ref>, we have r^⋆(_qf_p,b,c,δ) is the smallest positive root of the equation _qW_p,b,c,δ'(z)=0. That is, the radius of starlikeness of the normalized generalized Struve function _qf_p,b,c,δ(z) corresponds to the radius of starlikeness of the function _qΞ_p,b,c,δ(z)=_qW_p,b,c,δ'(z). The infinite series representations of the function _qΞ_p,b,c,δ(z) and its derivative are given as

    _qΞ_p,b,c,δ(z)=∑_n≥ 0(-1)^nc^n(2n+p+1)/2^2n+p+1n!Γ(qn+p/δ+b+2/2)z^2n+p

and

    _qΞ_p,b,c,δ'(z)=∑_n≥ 0(-1)^nc^n(2n+p)(2n+p+1)/2^2n+p+1n!Γ(qn+p/δ+b+2/2)z^2n+p-1.

Because of the facts that the function  _qW_p,b,c,δ is of the Laguerre-Pólya class of entire functions and that the class ℒ𝒫 is closed under differentiation, we deduce that the function _qΞ_p,b,c,δ is also in the class ℒ𝒫. This means that the zeros of the functions _qΞ_p,b,c,δ are all real. If we denote of the zeros of the function _qΞ_p,b,c,δ by _qε_p,b,c,δ,n, then the infinite product representation of the function _qΞ_p,b,c,δ is given as follows:

    2^p+1Γ(p/δ+b+2/2)_qΞ_p,b,c,δ(z)=(p+1)z^p∏_n≥1(1-z^2/_qε_p,b,c,δ,n^2).

With the help of the logarithmic derivation of the last equality, we obtain

    _qΞ_p,b,c,δ'(z)/_qΞ_p,b,c,δ(z)-p/z=-2∑_n≥ 1z/_qε_p,b,c,δ,n^2-z^2=-2∑_n≥ 1∑_k≥0z^2k+1/_qε_p,b,c,δ,n^2k+2=-2∑_k≥0τ_k+1z^2k+1,   | z|<_qε_p,b,c,δ,1,

where τ_k=∑_n≥ 1_qε_p,b,c,δ,n^-2k. On the other hand, by making use of the infinite sum representation of the function _qΞ_p,b,c,δ we arrive at

    _qΞ_p,b,c,δ'(z)/_qΞ_p,b,c,δ(z)=∑_n≥ 0(-1)^nc^n(2n+p)(2n+p+1)/2^2n+p+1n!Γ(qn+p/δ+b+2/2)z^2n+p-1/  ∑_n≥ 0(-1)^nc^n(2n+p+1)/2^2n+p+1n!Γ(qn+p/δ+b+2/2)z^2n+p.

By comparing the coefficients of the last two equalities, we have

    τ_1=c(p+3)Γ(p/δ+b+2/2)/4(p+1)Γ(q+p/δ+b+2/2) and  τ_2=c^2(p+3)^2Γ^2(p/δ+b+2/2)/16(p+1)^2Γ^2(q+p/δ+b+2/2)-c^2(p+5)Γ(p/δ+b+2/2)/16(p+1)Γ(2q+p/δ+b+2/2).

Now, by using the Euler-Rayleigh inequalities τ_k^-1/k<_qε_p,b,c,δ,1^2<τ_k/τ_k+1 for and k∈ℕ, we get the following inequality:

    2√((p+1)Γ(q+p/δ+b+2/2)/c(p+3)Γ(p/δ+b+2/2))<r^⋆(_qf_p,b,c,δ)<2√((p+1)(p+3)Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{(p+3)^2Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-(p+1)(p+5)Γ^2(q+p/δ+b+2/2)}).

No doubt we can find more tighter bounds for other values of k∈ℕ.


Let δ,q,b,c>0 and p+1>0. The radius of starlikeness r^⋆(_qg_p,b,c,δ) is satisfies

    2√(Γ(q+p/δ+b+2/2)/3cΓ(p/δ+b+2/2))<r^⋆(_qg_p,b,c,δ)<2√(3Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c(9Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-5Γ^2(q+p/δ+b+2/2))).



Our aim is to give more tight bounds for the radius of starlikeness r^⋆(_qg_p,b,c,δ) by making use of the Euler-Rayleigh inequalities. Let us recall that the radius of starlikeness of the function _qg_p,b,c,δ(z) is the first positive zero of its derivative, according to <cit.>. We can draw conclusion from <Ref> that the zeros of

    _qg_p,b,c,δ(z)=Γ(pδ+b+22)∑_n≥0(-1)^n c^n/n!Γ(qn+pδ+b+22)(z/2)^2n+1

all are real when δ,q,b,c>0 and p+1>0. Consequently, this function belongs to the Laguerre-Pólya class ℒ𝒫 of real entire functions (see <cit.> for more details), which are uniform limits of real polynomials whose all zeros are real. Now, since the Laguerre-Pólya class  ℒ𝒫 is closed under differentiation, it follows that _qg_p,b,c,δ' belongs also to the Laguerre-Pólya class and hence all of its zeros are real. Now, we consider the entire function

    _qφ_p,b,c,δ(z)=_qg_p,b,c,δ'(2√(z))=Γ(pδ+b+22)∑_n≥0(-1)^n(c)^n(2n+1)/n!Γ(qn+pδ+b+22)z^n .

Therefore in light of Laguerre's Lemma in <cit.> we obtain that the entire function

    _qγ_p,b,c,δ(z)=Γ(pδ+b+22)∑_n≥0c^n(2n+1)z^n  /n!Γ(qn+pδ+b+22)

has real zeros. Further, it is evident that the function has negative zeros since the coefficients of _qγ_p,b,c,δ are all positive. Hence _qγ_p,b,c,δ(-z) have real and positive zeros. That is to say, the function _qφ_p,b,c,δ(z) has real and only positive zeros. Suppose that _qα_p,b,c,δ,n's are  the zeros of the function _qφ_p,b,c,δ(z). Thus, since the function z↦_qφ_p,b,c,δ(z)=_qg_p,b,c,δ'(2√(z)) has growth order 11+q it can be represented by the product

    _qφ_p,b,c,δ(z)=∏_n≥ 1(1-z/_qα_p,b,c,δ,n)

where _qα_p,b,c,δ,n>0 for each n ∈ℕ. Now, by using the Euler-Rayleigh sum ℓ_k=∑_n ≥ 1_qα_p,b,c,δ,n^-k we obtain that

    _qφ_p,b,c,δ'(z)/_qφ_p,b,c,δ(z)=-∑_n ≥ 11/_qα_p,b,c,δ,n-z=-∑_n ≥ 1∑_k≥ 0z^k/_qα_p,b,c,δ,n^k+1=-∑_k≥ 0ℓ_k+1z^k,   | z| <_qα_p,b,c,δ,1,



    _qφ_p,b,c,δ'(z)/_qφ_p,b,c,δ(z)=∑_n≥ 0(-1)^n+1c^n+1(2n+3)/n!Γ(q(n+1)+pδ+b+22)z^n/ ∑_n≥ 0(-1)^n c^n (2n+1) /n!Γ(qn+pδ+b+22)z^n .

Equations (<ref>) and (<ref>) enable us to express the Euler-Rayleigh sums in terms of p,b,c,δ and by using the
Euler-Rayleigh inequalities ℓ_k^-1/k<_qα_p,b,c,δ,1<ℓ_k/ℓ_k+1 we get the inequalities for δ,q,b,c>0, p+1>0 and k∈ℕ

    2√(ℓ_k^-1/k)<r^⋆(_qg_p,b,c,δ)<2√(ℓ_k/ℓ_k+1).

Since

    ℓ_1=3cΓ(p/δ+b+2/2)/Γ(q+p/δ+b+2/2) and  ℓ_2=9c^2Γ^2(p/δ+b+2/2)/Γ^2(q+p/δ+b+2/2)-5c^2Γ(p/δ+b+2/2)/Γ(2q+p/δ+b+2/2)

in particular, for k=1 from the above Euler-Rayleigh inequalities we have the next inequality for 2√(_qα_p,b,c,δ,1), that is,

    2√(Γ(q+p/δ+b+2/2)/3cΓ(p/δ+b+2/2))<r^⋆(_qg_p,b,c,δ)<2√(3Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c(9Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-5Γ^2(q+p/δ+b+2/2))).

Of course, we can obtain more tighter bounds for other values of k∈ℕ.


The radius of starlikeness r^⋆(_qh_p,b,c,δ) is satisfies

    2Γ(q+p/δ+b+2/2)/cΓ(p/δ+b+2/2)<r^⋆(_qh_p,b,c,δ)<8Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{4Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-3Γ^2(q+p/δ+b+2/2) } .




Consider the infinite sum representation of _qh_p,b,c,δ(z) and its derivative

    _qh_p,b,c,δ(z)= Γ(p/δ+b+2/2)∑_n≥ 0(-1)^n c^n/4^n n!Γ(qn+p/δ+b+2/2)z^n+1,


    _qΥ_p,b,c,δ(z)=_qh_p,b,c,δ'(4z)=Γ(p/δ+b+2/2)∑_n≥ 0(-1)^n c^n(n+1)/ n!Γ(qn+p/δ+b+2/2)z^n .

Moreover, it is possible to prove the reality of the zeros of the function _qh_p,b,c,δ by using a similar approaching to the proof of <Ref>. This means that  _qh_p,b,c,δ belongs to the Laguerre-Pólya class ℒ𝒫. Therefore the function _qh_p,b,c,δ' belongs to the Laguerre-Pólya class ℒ𝒫 and has only real zeros. No doubt this is also valid for the function _qΥ_p,b,c,δ. That is to say, the function _qΥ_p,b,c,δ is the element of the Laguerre-Pólya class ℒ𝒫. Therefore, we can reach the conclusion from Laguerre's Lemma stated in <cit.> that the function _qΥ_p,b,c,δ has only positive zeros and has growth order 11+q, and thus _qΥ_p,b,c,δ(z) can be represented by the product

    _qΥ_p,b,c,δ(z)=∏_n≥ 1(1-z/_qς_p,b,c,δ,n),

where _qς_p,b,c,δ,n's are the zeros of the function _qΥ_p,b,c,δ(z) and _qς_p,b,c,δ,n>0 for each n∈ℕ. Now, by using the Euler-Rayleigh sum κ_k=∑_n ≥ 1_qς_p,b,c,δ,n^-k the infinite sum representation of the generalized Struve function _qW_p,b,c,δ we get

    _qΥ_p,b,c,δ'(z)/_qΥ_p,b,c,δ(z)=-∑_n ≥ 11/_qς_p,b,c,δ,n-z =-∑_n ≥ 1∑_k≥ 0z^k/_qς_p,b,c,δ,n^k+1=-∑_k≥ 0κ_k+1z^k,   | z|<_qς_p,b,c,δ,1,


    _qΥ_p,b,c,δ'(z)/_qΥ_p,b,c,δ(z)= ∑_n≥ 0(-1)^n+1c^n+1(n+2)/n!Γ(q(n+1)+p/δ+b+2/2)z^n / ∑_n≥ 0(-1)^n c^n(n+1)/n!Γ(qn+p/δ+b+2/2)z^n.

With the help of the above-mentioned last two equality we can express the Euler-Rayleigh sums in terms of p,b,c,δ and by using the
Euler-Rayleigh inequalities κ_k^-1/k<_qς_p,b,c,δ,1<κ_k/κ_k+1 we get the inequalities for 4_qς_p,b,c,δ,1 for δ,q,b,c>0, p+1>0 and k∈ℕ

    4κ_k^-1/k<r^⋆(_qh_p,b,c,δ)<4κ_k/κ_k+1.

Since

    κ_1=2cΓ(p/δ+b+2/2)/Γ(q+p/δ+b+2/2) and  κ_2=4c^2Γ^2(p/δ+b+2/2)/Γ^2(q+p/δ+b+2/2)-3c^2Γ(p/δ+b+2/2)/Γ(2q+p/δ+b+2/2)

in particular, for k=1 from the above Euler-Rayleigh inequalities we have the next inequality for  4_qς_p,b,c,δ,1, that is,

    2Γ(q+p/δ+b+2/2)/cΓ(p/δ+b+2/2)<r^⋆(_qh_p,b,c,δ)<8Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{4Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-3Γ^2(q+p/δ+b+2/2) },

of course we can obtain more tighter bounds for other values of k∈ℕ.



 §.§ The radii of convexity of order α of the functions _qf_p,b,c,δ, _qg_p,b,c,δ and _qh_p,b,c,δ.
 In this section we aim to determine the radii of convexity of the normalized generalized Struve functions and to find tight lower and upper bounds for the radius of convexity of these normalized Struve functions with the help of Euler-Rayleigh inequalities. 


Let δ,q,b,c>0,  p+1>0  and  α∈[0,1 ). Then the following assertions hold true.

  a. The radius of convexity of order α of the function _qf_p,b,c,δ is the smallest root of the equation

    1+( 1/p+1-1)r_qW_p,b,c,δ'(r)/_qW_p,b,c,δ(r)+r_qW_p,b,c,δ”(r)/_qW_p,b,c,δ'(r)=α.


  b. The radius of convexity of order α of the function _qg_p,b,c,δ is the smallest root of the equation

    1+r_qg_p,b,c,δ”(r)/_qg_p,b,c,δ'(r)=α.



  c. The radius of convexity of order α of the function _qh_p,b,c,δ is the smallest root of the equation

    1+z_qh_p,b,c,δ”(z)/_qh_p,b,c,δ'(z)=α.




  a.
It is obvious that 

    1+ z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z)=1+( 1/p+1-1)z_qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z)+z_qW_p,b,c,δ”(z)/_qW_p,b,c,δ'(z).

Now, by making use of <Ref> we can write the following infinite product representations

    2^p+1Γ(p/δ+b+2/2)_qW_p,b,c,δ(z)   =z^p+1∏_n≥ 1(1-z^2/_qω_p,b,c,δ,n^2), 
    
    2^p+1Γ(p/δ+b+2/2)_qW_p,b,c,δ'(z)   =(p+1)z^p∏_n≥ 1(1-z^2/_qω_p,b,c,δ,n'^2)

where _qω_p,b,c,δ,n and _qω_p,b,c,δ,n' stand for the nth positive roots of _qW_p,b,c,δ and _qW_p,b,c,δ', respectively. Logarithmic differentiation of the above equality leads to

    z_qW_p,b,c,δ'(z)/_qW_p,b,c,δ(z)= p+1-∑_n ≥ 12z^2/_qω_p,b,c,δ,n^2-z^2,    z_qW_p,b,c,δ”(z)/_qW_p,b,c,δ'(z)=p-∑_n ≥ 12z^2/_qω_p,b,c,δ,n'^2-z^2,

which implies that 

    1+z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z)=1-(1/p+1-1)∑_n≥12z^2/_qω_p,b,c,δ,n^2-z^2-∑_n ≥ 12z^2/_qω_p,b,c,δ,n'^2-z^2.

We will prove the theorem in two steps. First suppose p∈(-1,0 ] . By using the inequality (<ref>), for all z∈𝔻(0,_qω_p,b,c,n') it is easy to deduce the following inequality

    (1+z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z))≥1-(1/p+1-1)∑_n≥12r^2/_qω_p,b,c,δ,n^2-r^2-∑_n ≥ 12r^2/_qω_p,b,c,δ,n'^2-r^2,

where | z|=r. In the second step, it is easy to see that if we use the inequality (<ref>) then we conclude that the above inequality
is also fulfilled when p>0 . Here we used that the zeros _qω_p,b,c,δ,n and _qω_p,b,c,δ,n' interlace according to <Ref>. Now, the above deduced inequality implies for r∈(0, _qω_p,b,c,δ,1')

    inf_z∈𝔻_r{(1+z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z))}=1+r_qf_p,b,c,δ”(r)/_qf_p,b,c,δ'(r).

Now we deal with the function _qϑ_p,b,c,δ:(0, _qω_p,b,c,δ,1' ) →ℝ, defined by

    _qϑ_p,b,c,δ(r)=1+r_qf_p,b,c,δ”(r)/_qf_p,b,c,δ'(r).

The function is strictly decreasing since

    _qϑ_p,b,c,δ'(r)   =-(1/p+1-1 )∑_n ≥ 14r_qω_p,b,c,δ,n^2/(_qω_p,b,c,δ,n^2-r^2) ^2-∑_n≥14r_qω_p,b,c,δ,n'^2/(_qω_p,b,c,δ,n'^2-r^2)^2
       <∑_n≥14r_qω_p,b,c,δ,n^2/(_qω_p,b,c,δ,n^2-r^2)^2-∑_n≥14r_qω_p,b,c,δ,n'^2/(_qω_p,b,c,δ,n'^2-r^2)^2<0

for p+1>0 and r∈(0, _qω_p,b,c,δ,1' ). It is important to note that here we used again that the zeros _qω_p,b,c,δ,n and _qω_p,b,c,δ,n' interlace and with conditions of <Ref> we get that

    _qω_p,b,c,δ,n^2( _qω_p,b,c,δ,1'^2-r^2)^2<_qω_p,b,c,δ,n'^2( _qω_p,b,c,δ,1^2-r^2)^2.

Also, taking into consideration that lim_r ↘ 0_qϑ_p,b,c,δ(r)=1-α>0, lim_r↗_qω_p,b,c,δ,1_qϑ_p,b,c,δ(r)=-∞ that means that for z∈𝔻(0,r_1) we have

    (1+z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z)) >α,

if and only if r_1 is the unique root of

    1+z_qf_p,b,c,δ”(z)/_qf_p,b,c,δ'(z)=α,

situated in (0, _qω_p,b,c,δ,1' ).

  b. The definition of the function _qg_p,b,c,δ implies that this function is of the Laguerre-Pólya class of entire functions. That is, the function _qg_p,b,c,δ is entire function that has only real zeros. Suppose that _qβ_p,b,c,δ,n's are the real zeros of the function _qg_p,b,c,δ.  By making use of (<ref>) we conclude

    1+z_qg_p,b,c,δ”(z)/_qg_p,b,c,δ'(z)=1-∑_n ≥ 12z^2/_qβ_p,b,c,δ,n^2-z^2.

By applying the inequality (<ref>) we obtain

    (1+z_qg_p,b,c,δ”(z)/_qg_p,b,c,δ'(z)) ≥ 1- ∑_n ≥ 12r^2/_qβ_p,b,c,δ,n^2-r^2,

where |z |=r. Whence for r∈(0,_qβ_p,b,c,δ,1) we have

    inf_z∈𝔻_r{(1+z_qg_p,b,c,δ”(z)/_qg_p,b,c,δ'(z))}= 1- ∑_n ≥ 12r^2/_qβ_p,b,c,δ,n^2-r^2=1+r_qg_p,b,c,δ”(r)/_qg_p,b,c,δ'(r).

The function _qϕ_p,b,c,δ:(0,_qβ_p,b,c,δ,1)→ℝ defined by

    _qϕ_p,b,c,δ(r)=1+r_qg_p,b,c,δ”(r)/_qg_p,b,c,δ'(r),

is strictly decreasing and 

    lim_r↗_qβ_p,b,c,δ,1_qϕ_p,b,c,δ(r)=-∞,   lim_r ↘ 0_qϕ_p,b,c,δ(r)=1.

As a result, the equation

    1+r_qg_p,b,c,δ”(r)/_qg_p,b,c,δ'(r)=α

has a unique root r_2 situated in (0,_qβ_p,b,c,δ,1).

  c. It is obvious that 

    1+z_qh_p,b,c,δ”(z)/_qh_p,b,c,δ'(z)=1-∑_n ≥ 1z/_qθ_p,b,c,δ,n^2-z.

Let r ∈ (0, _qθ_p,b,c,δ,1^2) be a fixed number. Because of the minimum principle for harmonic functions and inequality (<ref>) for λ=0 we get

    (1+z_qh_p,b,c,δ”(z)/_qh_p,b,c,δ'(z))=1-(∑_n≥1z/_qθ_p,b,c,δ,n^2-z) ≥ 1+r_qh_p,b,c,δ”(r)/_qh_p,b,c,δ'(r).

Consequently, it follows that

    inf_z∈𝔻_r{( 1+z_qh_p,b,c,δ”(z)/_qh_p,b,c,δ'(z)) }=1+r_qh_p,b,c,δ”(r)/_qh_p,b,c,δ'(r).

The function _qΘ_p,b,c,δ:(0,_qθ_p,b,c,δ,1^2)→ℝ defined by

    _qΘ_p,b,c,δ(r)= 1+r_qh_p,b,c,δ”(r)/_qh_p,b,c,δ'(r)

is strictly decreasing and 

    lim_r↗_qθ_p,b,c,δ,1^2_qΘ_p,b,c,δ(r)=-∞,   lim_r ↘ 0_qΘ_p,b,c,δ(r)=1.

Consequently, the equation

    1+z_qh_p,b,c,δ”(z)/_qh_p,b,c,δ'(z)=α

has a unique root r_3 in (0,_qθ_p,b,c,δ,1^2).

The proof is completed.

The following theorems are related to some tight lower and upper bounds for the radii of convexity of the normalized generalized Struve functions.

Let δ,q,b,c>0 and p+1>0. Then the radius of convexity r^c(_qg_p,b,c,δ) of the function 

    z↦_qg_p,b,c,δ(z)=2^p+1Γ(p/δ+b+2/2)z^-p_qW_p,b,c,δ(z),

 is the smallest root of the (z_qg_p,b,c,δ')^'=0 and satisfies the following inequality

    2/3√(Γ(q+p/δ+b+2/2)/cΓ(p/δ+b+2/2))<r^c(_qg_p,b,c,δ)<6√(Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-25Γ^2(p/q+δ+b+2/2) }).



For proving our main result we will need the Alexander’s duality theorem which has a very simple proof based on the characterization of starlike and convex functions in the unit disc. Owing to this theorem one can deduce that the function _qg_p,b,c,δ(z) is convex if and only if z↦( z_qg_p,b,c,δ)^' is starlike. From the studies in <cit.> we know that the smallest positive zero of z↦(z_qg_p,b,c,δ')^' is the radius of starlikeness of z_qg_p,b,c,δ'(z). That is why the radius of convexity r^c(_qg_p,b,c,δ) is the smallest positive root of the equation (z_qg_p,b,c,δ')^'=0. Now, by using the infinite series representations of the generalized Struve fuction and its derivative

    _qW_p,b,c,δ(z)   =∑_n≥ 0(-1)^nc^nz^2n+p+1/2^2n+p+1n!Γ(qn+p/δ+b+2/2), 
    _qW_p,b,c,δ'(z)   =∑_n≥ 0(-1)^nc^n(2n+p+1)z^2n+p/2^2n+p+1n!Γ(qn+p/δ+b+2/2),

respectively, we arrive at 

    _qΔ_p,b,c,δ(z)=(z_qg_p,b,c,δ'(z) )'=Γ(pδ+b+22)∑_n≥ 0(-1)^nc^n(2n+1)^2/2^2nn!Γ(qn+p/δ+b+2/2)z^2n.

Moreover, it is obvious that

    _qΛ_p,b,c,δ(z)=_qΔ_p,b,c,δ(2√(z))=Γ(pδ+b+22)∑_n≥ 0(-1)^nc^n(2n+1)^2/n!Γ(qn+p/δ+b+2/2)z^n.

Taking into account facts that the function _qg_p,b,c,δ belongs to the Laguerre-Pólya class of entire functions and that the class ℒ𝒫 is closed under differentiation, it is easy to deduce that the function _qΛ_p,b,c,δ belongs to the Laguerre-Pólya class. As a result, the function _qΛ_p,b,c,δ is an entire function that has only real zeros. In this case, we can continue to prove the theorem by emulating the steps taken in the proof of <Ref>.  Suppose that _qϱ_p,b,c,δ,n's are the positive zeros of the function _qΛ_p,b,c,δ. Then the function _qΛ_p,b,c,δ has the infinite product representation as follows:

    _qΛ_p,b,c,δ(z)=∏_n≥ 1( 1-z/_qϱ_p,b,c,δ,n).

Having taken the logarithmic derivation of the above equality, we obtain

    _qΛ_p,b,c,δ'(z)/_qΛ_p,b,c,δ(z)=-∑_n≥ 11/_qϱ_p,b,c,δ,n-z=-∑_k≥0μ_k+1z^k,  |z |< _qϱ_p,b,c,δ,1,

where μ_k=∑_n≥ 1_qϱ_p,b,c,δ,n^-k. Furthermore, by making use of the function _qΛ_p,b,c,δ we obtain

    _qΛ_p,b,c,δ'(z)/_qΛ_p,b,c,δ(z)=∑_n≥ 0(-1)^nc^n(2n+1)^2/n!Γ(qn+p/δ+b+2/2)z^n/ ∑_n≥ 0(-1)^n+1c^n+1(2n+3)^2/n!Γ(qn+1+p/δ+b+2/2)z^n.

It is evident that equations (<ref>) and (<ref>) enable us to express the Euler-Rayleigh sums in terms of q,p,b,c,δ and by considering the  Euler-Rayleigh inequalities μ_k^-1/k<_qϱ_p,b,c,δ,1<μ_k/μ_k+1 we obtain the inequalities for 2√(_qϱ_p,b,c,δ,1)  for δ,q,b,c>0, p+1>0 and k∈ℕ

    2√(μ_k^-1/k)<r^c(_qg_p,b,c,δ)<2√(μ_k/μ_k+1).

As a result of the comparison of the coefficients of (<ref>) and (<ref>), we arrive at

    μ_1=9cΓ(pδ+b+22)/Γ(q+pδ+b+22) and  μ_2=81c^2Γ^2(pδ+b+22)/Γ^2(q+pδ+b+22)-25c^2Γ(pδ+b+22)/Γ(2q+pδ+b+22).

Obviously, for k=1 from the Euler-Rayleigh inequalities, It can be observed that the following inequality come to light

    2/3√(Γ(q+p/δ+b+2/2)/cΓ(p/δ+b+2/2))<r^c(_qg_p,b,c,δ)<6√(Γ(q+p/δ+b+2/2)Γ(2q+p/δ+b+2/2)/c{Γ(p/δ+b+2/2)Γ(2q+p/δ+b+2/2)-25Γ^2(p/q+δ+b+2/2) }).

No doubt it can be presented more tighter bounds for other values k∈ℕ.
	


Let δ,q,b,c>0 and p+1>0. Then the radius of convexity r^c(_qh_p,b,c,δ) of the function 

    z↦_qh_p,b,c,δ(z)= 2^p+1Γ(p/δ+b+2/2)z^1-p+1/2_qW_p,b,c,δ(√(z))

is the smallest root of the (z_qh_p,b,c,δ'(z))^'=0 and satisfies the following inequality

    Γ(q+pδ+b+22)/cΓ(pδ+b+22)<r^c(_qh_p,b,c,δ)<16Γ(q+pδ+b+22)Γ(2q+pδ+b+22)/c{16Γ(pδ+b+22)Γ(2q+pδ+b+22)-9Γ^2(q+pδ+b+22) }.



In light of explanations which have been presented in the proof of previous theorem (that is, <ref>) one can deduce that the radius of convexity r^c(_qh_p,b,c,δ) is the smallest positive root of the equation (z_qh_p,b,c,δ'(z))^'=0. By taking √(z) instead of z in (<ref>) and (<ref>), respectively we can draw a conclusion

    _qλ_p,b,c,δ(z)=(z_qh_p,b,c,δ'(z))^'=1+Γ(pδ+b+22)∑_n≥ 1(-1)^nc^n(n+1)^2/2^2nn!Γ(qn+p/δ+b+2/2)z^n.

With the help of facts that the function _qh_p,b,c,δ belongs to the Laguerre-Pólya class of entire functions ℒ𝒫 and that the class  ℒ𝒫 is closed under differentiation, it is not hard to say that the function _qλ_p,b,c,δ is also in the Laguerre-Pólya class. This means that the zeros of the function _qλ_p,b,c,δ are all real. Furthermore, by means of approach used in the proof of <ref> we can say that all of its zeros are positive.  Suppose that  _qτ_p,b,c,δ,ns are the zeros of the function _qλ_p,b,c,δ. Then the infinite product representation of the function _qλ_p,b,c,δ can be given as

    _qλ_p,b,c,δ(z)=∏_n≥ 1(1-z/_qτ_p,b,c,δ,n).

Having taken the logarithmic derivation of Eq. (<ref>), we get

    _qλ_p,b,c,δ'(z)/_qλ_p,b,c,δ(z)=-∑_k≥ 0υ_k+1z^k,   |z |<_qτ_p,b,c,δ,1,

where υ_k=∑_n ≥ 1_qτ_p,b,c,δ,n^-k. Also, by making use of the derivative of Eq. (<ref>), which is related to infinite sum representation of _qλ_p,b,c,δ(z), we obtain

    _qλ_p,b,c,δ'(z)/_qλ_p,b,c,δ(z)=∑_n≥0(-1)^n+1c^n+1(n+2)^2/2^2n+2n!Γ(q(n+1)+p/δ+b+2/2)z^n/ ∑_n≥ 0(-1)^nc^n(n+1)^2/2^2nn!Γ(qn+p/δ+b+2/2)z^n.

The outcomes found by comparing the coefficients of Eq. (<ref>) and (<ref>) can be given as follows:

    υ_1=cΓ(pδ+b+22)/Γ(q+pδ+b+22),   υ_2=c^2Γ^2(pδ+b+22)/Γ^2(q+pδ+b+22)-9c^2Γ(pδ+b+22)/16Γ(2q+pδ+b+22).

By considering the Euler-Rayleigh inequalities υ_k^-1/k<_qτ_p,b,c,δ,1<υ_k/υ_k+1 for δ,q,b,c>0, p+1>0 and k=1 we have

    Γ(q+pδ+b+22)/cΓ(pδ+b+22)<r^c(_qh_p,b,c,δ)<16Γ(q+pδ+b+22)Γ(2q+pδ+b+22)/c{16Γ(pδ+b+22)Γ(2q+pδ+b+22)-9Γ^2(q+pδ+b+22) }.

Obviously, it can be presented more tighter bounds for other values k∈ℕ.



 §.§ Some particular cases of the main results

This section is devoted to the some intriguing results which are obtained by making some comparisons with earlier results. It is possible to say that generalized Struve function is actually a generalization of the suitible transformation of the Bessel function of the first kind. That is, we have the relation

    _1W_ν-1,2,1,1(z)=J_ν(z)

where J_ν is the Bessel function of the first kind and order ν. By considering this relation, we see that our main results obtained in this paper coincide with the following listed studies. 

It is evident that our main results which are given in <ref>, in particular for q=2, p=ν-1, b=2, c=1  δ=1, correspond to the results in <cit.>.

Let ν>-1  and  α∈[0,1 ).

	
  a. The radius of starlikeness of order α of _1f_ν-1,2,1,1(z)=[2^νΓ(ν+1)J_ν(z)]^1/ν is the smallest positive root of the equation
	
    zJ_ν'(z)-αν J_ν(z)=0.

	
  b. The radius of starlikeness of order α of _1g_ν-1,2,1,1(z)=2^νΓ(ν+1)z^1-νJ_ν(z) is the smallest positive root of the equation
	
    zJ_ν'(z)+(1-α-ν)J_ν(z)=0.

	
  c. The radius of starlikeness of order α of _1h_ν-1,2,1,1(z)=2^νΓ(ν+1)z^1-ν/2J_ν(√(z)) is the smallest positive root of the equation
	
    zJ_ν'(z)+(2-2α-ν)J_ν(z)=0.

 


By choosing the values q=2, p=ν-1, b=2, c=1  δ=1 in <ref> we obtain the following corollary which is new.

If ν>-1, then the radius of starlikeness r^⋆(_1f_ν-1,2,1,1) satisfies

    2√(ν(ν+1)/ν+2)<r^⋆(_1f_ν-1,2,1,1)<2(ν+2)√(ν(ν+1)/ν^2+8ν+8).



It is obvious that our main results which are presented in <ref> and <ref> when we take q=2, p=ν-1, b=2, c=1  δ=1, coincide with the inequalities in <cit.> and <cit.>, respectively.

Let ν>-1. The following assertions hold true:

	
  a. The radius of starlikeness r^⋆(_1g_ν-1,2,1,1) satisfies
	
    2√(ν+1/3)<r^⋆(_1g_ν-1,2,1,1)<2√(3(ν+1)(ν+2)/4ν+13) .

	
  b. The radius of starlikeness r^⋆(_1h_ν-1,2,1,1) satisfies
	
    2(ν+1)<r^⋆(_1h_ν-1,2,1,1)<8(ν+1)(ν+2)/ν+5 .




It is clear that  our main results which are given in  <ref> when we choose q=2, p=ν-1, b=2, c=1  δ=1, correspond to the results given in <cit.>, <cit.> and <cit.>.

Let ν>-1  and  α∈[0,1 ). Then the following assertions hold true

	
  a. The radius of convexity of order α of the function _1f_ν-1,2,1,1 is the smallest positive root of the equation
	
    1+rJ_ν”(r)/J_ν'(r)+(1/ν-1 )rJ_ν'(r)/J_ν(r)=α.

	
  b. The radius of convexity of order α of the function _1g_ν-1,2,1,1 is the smallest positive root of the equation
	
    1+rJ_ν+2(r)-3J_ν+1(r)/J_ν(r)-rJ_ν+1(r)=α.

	Here we used the recurrence formula zJ_ν'(z)=ν J_ν(z)-zJ_ν+1(z).
	
  c. The radius of convexity of order α of the function _1h_ν-1,2,1,1 is the smallest positive root of the equation
	
    1+r^12/2.r^12J_ν+2(r^12)-4J_ν+1(r^12)/2J_ν(r^12)-r^12J_ν+1(r^12)=α.




It is clear that  our main results which are given in <ref> and <ref> when we choose q=2, p=ν-1, b=2, c=1  δ=1, tally with the results given in <cit.> and <cit.>, respectively.

Let ν>-1. The following assertions hold true:

	
  a. The radius of convexity r^c(_1g_ν-1,2,1,1) satisfies the following inequality:
	
    2√(ν+1)/3<r^c(_1g_ν-1,2,1,1)<6√((ν+1)(ν+2)/56ν+137).

	
  b. The radius of convexity r^c(_1h_ν-1,2,1,1) satisfies the following inequality:
	
    ν+1 < r^c(_1h_ν-1,2,1,1)<16(ν+1)(ν+2)/7ν+23.





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