On Bernstein's inequality for entire functions of exponential type Export

@article{citeulike:4750183, abstract = {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.}, author = {Rahman, Q. I. and Tariq, Q. M.}, citeulike-article-id = {4750183}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.jmaa.2009.05.035}, citeulike-linkout-1 = {http://linkinghub.elsevier.com/retrieve/pii/S0022247X09004478}, day = {01}, doi = {10.1016/j.jmaa.2009.05.035}, issn = {0022247X}, journal = {Journal of Mathematical Analysis and Applications}, keywords = {complex\_analysis, inequalities, univariate\_functions}, month = {November}, number = {1}, pages = {168--180}, posted-at = {2009-07-11 14:21:46}, priority = {3}, title = {On Bernstein's inequality for entire functions of exponential type}, url = {http://dx.doi.org/10.1016/j.jmaa.2009.05.035}, volume = {359}, year = {2009} }

TY - JOUR ID - citeulike:4750183 L3 - citeulike-article-id:4750183 N2 - 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. IS - 1 JF - Journal of Mathematical Analysis and Applications SN - 0022247X EP - 180 TI - On Bernstein's inequality for entire functions of exponential type VL - 359 SP - 168 KW - complex_analysis KW - inequalities KW - univariate_functions AU - Rahman, QI AU - Tariq, QM PY - 2009/11/01/ UR - http://dx.doi.org/10.1016/j.jmaa.2009.05.035 ER -