Some Operator Preserving Inequalities Involving Functions of Exponential Type

Abstract

Let f be an entire function of exponential type τ such that |f(x)| ≤ M on the real axis. Then, it was proved by Bernstein [8, Theorem 14.1.7] that |f′(x)| ≤ M τ.
In the literature, there exist several results estimating |f′(x)| in terms of sup_-∞<x<∞|f(x)|.
Here we prove some Bernstein-type inequalities for an operator L defined by L[f(z)] =e^βz[e^−βzf(z)]′, where β is any complex number. These inequalities generalise some well-known inequalities for entire functions of exponential type.

Key words and phrases. Functions of exponential type, Asymmetric Functions, B-Operators.

2010 Mathematics Subject Classification. Primary: 30A10, 30C10, 30D15; Secondary: 42A05