Extremal problems in a subclass of entire functions of finite power

S. A. Kas'yanyuk

Matematičeskie zametki, Tome 19 (1976) no. 1, pp. 19-28

Résumé

We study the subclass $W_\sigma(A)$ of the class of entire transcendental functions $f(z)$of exponential type with index not greater than sgr satisfying the condition $$ \int_{-\infty}^\infty|f(x)|^2\,dx\le A^2 $$ We find the set of values of the quantities $f(z)$, $f'(z)$, etc. when $z$ is fixed and $f(z)$ runs through the subclass $W_\sigma(A)$. We study extremal values of functionals of the type $\Phi(f(z),f'(z))$. In particular, we obtain upper bounds on the quantities $|f(z+\beta/2)\pm f(z-\beta/2)|$ и $|af'(z)+b\sigma f(z)|$.