On Certain Extremal Problems for Functions with Positive Real Part

Abstract

Abstract. For the class P of analytic functionsp(z),p(0) = 1, with positive real part in \z\ < 1, a type of extremal problems is determined which can be solved already within the setp(z) = (1 + ez)/(\ -ez), |e| = 1. One problem of this kind is to find the largest number p(s, p) such that Re{p(z) + szp'(z)/ (p(z) + n)} > 0, |z| < p(s, /i),for all p E P, -ljt(i£C,i>0.Sharp upper bounds for two other functionals over P are also given.

The Tilted Carathéodory Class and Its Applications

“…By using fundamental functional analysis, we obtain the sharp estimate of Re zp ′ (z)/p(z) with p(z) ∈ P λ . The estimate of this entity was considered by some authors, see Bernardi [2] and Robertson [25] , and the sharp one appeared in a paper of Ruscheweyh and Singh [29]. Although our estimate is equivalent to that in [29], the extremal functions which make the estimate sharp are also given explicitly.…”

“…The sharp estimate of the entity in Theorem 6 first appears in a paper [29] by Ruscheweyh and Singh. Their proof was based on variational method.…”

“…The Carathéodory class P 0 occupies an extremely important place in the theory of functions and has been studied by many authors ( [5,Chapter 7,p. 77], [6], [17], [19], [24], [25], [29], [30], [36], [37]). The tilted Carathéodory class P λ scatters in some papers (see [15], [16], [19], [26]), although the name was not given in the literature.…”