Abstract
By the help of power series f (z) =
Introduction
The numerical radius w(T) of an operator T on H is given by [, p.] w(T) = sup |λ|, λ ∈ W (T) = sup Tx, x , x =  .
(.)
Obviously, by (.), for any x ∈ H, one has
Tx, x ≤ w(T) x  .
(  .  )
It is well known that w(·) is a norm on the Banach algebra B(H) of all bounded linear operators T : H → H, i.e., (i) w(T) ≥  for any T ∈ B(H) and w(T) =  if and only if T = ;
(ii) w(λT) = |λ|w(T) for any λ ∈ C and T ∈ B(H);
(iii) w(T + V ) ≤ w(T) + w(V ) for any T, V ∈ B(H). http://www.journalofinequalitiesandapplications.com/content/2013/1/298
This norm is equivalent with the operator norm. In fact, the following more precise result holds [, p.].
Theorem  (Equivalent norm) For any T ∈ B(H), one has w(T) ≤ T ≤ w(T).
(.) Some improvements of (.) are as follows.
Theorem  (Kittaneh,  []) For any operator T ∈ B(H), we have the following refinement of the first inequality in (.):
Utilizing the Cartesian decomposition for operators, Kittaneh improved the inequality (.) as follows.
Theorem  (Kittaneh,  []) For any operator T ∈ B(H), we have
From a different perspective, we have the following result as well. Motivated by the above results, we establish in this paper some inequalities for the numerical radius of functions of operators defined by power series, which incorporate many fundamental functions of interest such as the exponential function, some trigonometric functions, the functions f (z) = ( ± z) - , g(z) = log( ± z) - and others. Some examples of interest are also provided.
Theorem  (Dragomir,  []) For any operator T ∈ B(H), we have
w  (T) ≤   w T  + T  . (  .  )
Some inequalities for one operator
Now, by the help of power series f (z) = ∞ n= a n z n , we can naturally construct another power series which will have as coefficients the absolute values of the coefficients of the original series, namely, f a (z) := ∞ n= |a n |z n . It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients a n ≥ , then f a = f . The following simple result provides some nice inequalities for operator functions defined by power series.
be a function defined by power series with complex coefficients and convergent on the open disk D(, R) ⊂ C, R > . For any T ∈ B(H), we have the inequality
provided w(T) < R, and the inequality
Proof Let m ∈ N with m ≥ . Utilizing the properties of the numerical radius, we have
Since the series ∞ n= |a n |w n (T) is convergent on R, the series ∞ n= a n T n is convergent in B(H), and by the continuity of the numerical radius, we have
Then, by letting m → ∞ in the inequality (.), we deduce the desired result (.).
Utilizing the properties of the numerical radius and the Kittaneh inequality (.), we also have
Since the series 
From a different perspective, we have the following.
be a function defined by power series with complex coefficients and convergent on the open disk D(, R) ⊂ C, R > . For any T ∈ B(H) with T  < R
and z ∈ C with |z|  < r, we have the inequality
and the inequality Now, on writing the inequality (.) for the operators T n , we have
On making use of the inequalities (.)-(.), we get we deduce the desired result (.). Now, on making use of the Kittaneh inequality (.), we also have
By the inequalities (.) and (.), we then get 
and the inequality
Some inequalities for two operators
We start with the following result.
Theorem  Let A, B ∈ B(H) and k >  such that w(AB) ≤ k  w(A)w(B). (.)

If f (z) = ∞ n= a n z n is a function defined by power series with complex coefficients and convergent on the open disk D(, R) ⊂ C, R >  and k
where
and
Proof By the properties of the numerical radius and by (.), we have
Let m ∈ N with m ≥ . We have, by the above inequality, 
that holds for nonnegative numbers m k and complex numbers x k , y k , where k ∈ {, . . . , n}, we observe that the convergence of the series
Utilizing (.), we can state that Since all the series whose partial sums are involved in (.) are convergent, then by taking n → ∞ in (.), we deduce the second inequality in (.).
Remark  If we take p = q =  in the first inequality in (.), we have where and
) .
If A, B ∈ B(H) are commutative with
where
and
) .
The proof of the inequality (.) follows by Theorem  since in this case, we can take k =  in (.), while the inequality (.) follows by the commutative case, in which case we can take k = √  in (.). The case of commuting operators can be treated in a different way as well. 
where
and
Proof Since A, B ∈ B(H) are commutative, then for any n ∈ N, the operators A n and B n are commutative and A n B n = (AB) n .
Applying Theorem  for the commutative case, we have
for any n ∈ N. Let m ∈ N with m ≥ . We have, by the above inequality, Now, on making use of a similar approach to the one employed in the proof of Theorem , we deduce the desired result (.).
Remark  If we take p = q =  in the first inequality in (.), we get
As pointed out in the introduction, the inequality 
Moreover, if the operators A and B double commute, then the constant c can be taken to be  in (.).
Proof Applying the inequality (.) for the commuting operators A n and B n with n ∈ N, we have w (AB) n = w A n B n ≤ cw A n B n ≤ cw n (A) B n for any n ∈ N. On making use of a similar argument as in the proof of Theorem , we deduce the desired inequality (.).
If the operators A and B double commute, then the operators A n and B n also double commute, and by Theorem  we deduce the second part of the proposition.
From a different perspective, we have the following result as well.

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