On Bernsteinʹs inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that | f ( x ) | ⩽ M for − ∞ < x < ∞ , then | f ′ ( x ) | ⩽ M τ for − ∞ < x < ∞ . If p is a polynomial of degree at most n with | p ( z ) | ⩽ M for | z | = 1 , then f ( z ) : = p ( e i z ) is an entire function of exponential type n with | f ( x ) | ⩽ M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, | p ′ ( z ) | ⩽ M n for | z | = 1 . Lately, many papers have been written on polynomials p that satisfy the condition z n p ( 1 / z ) ≡ p ( z ) . They do form an intriguing class. If a polynomial p satisfies this condition, then f ( z ) : = p ( e i z ) is an entire function of exponential type n that satisfies the condition f ( z ) ≡ e i n z f ( − z ) . This led Govil [N.K. Govil, L p inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445–452] to consider entire functions f of exponential type satisfying f ( z ) ≡ e i τ z f ( − z ) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.