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Coefficients estimates in subclasses of the Carathéodory class related to conical domains
"... Abstract. We study some properties of subclasses of of the Carathéodory class of functions, related to conic sections, and denoted by P(pk). Coefficients bounds, estimates of some functionals are given. 1. ..."
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Abstract. We study some properties of subclasses of of the Carathéodory class of functions, related to conic sections, and denoted by P(pk). Coefficients bounds, estimates of some functionals are given. 1.
SHARP NORM ESTIMATE OF SCHWARZIAN DERIVATIVE FOR A CLASS OF CONVEX FUNCTIONS
, 2011
"... We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157169]. As applications, we give sharp norm estimates for strongly conv ..."
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We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157169]. As applications, we give sharp norm estimates for strongly convex functions of order α, 0 < α < 1, and for uniformly convex functions.
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"... We explore a method which is implicit in a paper of Burkholder of identifying the H2 Hardy norm of a conformal map with the explicit solution of Dirichlet’s problem in the complex plane. Using the series form of the Hardy norm, we obtain an identity for the sum of a series obtained from the conforma ..."
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We explore a method which is implicit in a paper of Burkholder of identifying the H2 Hardy norm of a conformal map with the explicit solution of Dirichlet’s problem in the complex plane. Using the series form of the Hardy norm, we obtain an identity for the sum of a series obtained from the conformal map. We use this technique to evaluate several hypergeometric sums, as well as several sums that can be expressed as convolutions of the terms in a hypergeometric series. The most easily stated of the identities we obtain are Euler’s famous Basel sum, as well as the sum 12+