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.

Some Classes of k-Uniformly Functions with Bounded Radius Rotation

“…This implies, by using a result due to Miller-Mocann [8, p 113, Theorem 3.3 e] that h i ∈ P(σ ), where σ is given by (15). For sharpness, the extremal function is given as…”

“…As a special case, when n = 0, we note that R m (1, α) = V m (α), where V m (α) is the class of functions of bounded boundary rotation with order α, see [3], and f ∈ V m (α) implies that f ∈ R m (0, σ ) = R m (σ 0 ), where R m (σ 0 ) is the corresponding class of bounded radius rotation with order σ 0 , order σ 0 is given by (15) with n = 0, as follows.…”

On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation

“…(See[10].) Let p be an analytic function in E with p(0) = 1 and Re{p(z)} > 0, z ∈ E. Then, for s > 0 and μ = −1 (complex),…”