What is the mathematical relation between hfe and hie
Both quantities (\${h_{fe}}\$ and \${h_{ie}}\$) are linked to one another via the transconductance \${g_m}\$.

Derivation:

\${h_{fe}} = \frac{{{I_C}}}{{{I_B}}} = \frac{{d\left( {{I_C}} \right)}}{{d\left( {{I_B}} \right)}}\$,

with

\$d\left( {{I_B}} \right) = \frac{{d({V_{BE}})}}{{{h_{ie}}}}\$.

Therefore:

\$\frac{{{h_{fe}}}}{{{h_{ie}}}} = \frac{{d\left( {{I_C}} \right)}}{{d\left( {{V_{BE}}} \right)}} = {g_m}\$,

with

\${g_m} = \frac{{{I_C}}}{{{V_T}}}\$ (\${V_T}\$: temperature voltage)

(The transconductance \${g_m}\$ is identical to the slope of the \${I_C} = f\left(V_{BE}\right)\$ characteristics.)