Sanford S. Miller

Second order differential inequalities in the complex plane

Let w(z) be regular in the unit disc U and let h(r, s, t) be a complex function defined in a domain of C3. The authors determine conditions on h such that |h(w(z), zw′(z), z2w″(z))| 0 implies Re w(z) >0. Applications of these results to univalent function theory, differential equations and harmonic functions are given.

On a Briot-Bouquet Differential Subordination

Let p(z) be analytic in the unit disc Δ, let h(z) be convex (univalent) in Δ, and let β and γ be complex numbers. The authors show that if p(z) = zp′(z)(βp(z) + γ)−1 ≺ h(z) (where ≺ denotes subordination), then p(z) ≺ h(z). They prove, further, that if, in addition, the differential equation q(z) + zq′(z)(βq(z) + γ)−1 = h(z) has a univalent solution q(z), then the sharp subordination p(z) ≺ q(z) holds. Applications of these results in the field of univalent functions are given.

Differential subordinations and inequalities in the complex plane

Abstract Let p be analytic in the unit disc U and let q be univalent in U. In addition, let Ω be a set in C and let ψ: C3 × U → C. The authors determine conditions on ψ so that {ψ(p(z), zp′(z), z 2 p″(z);z)¦z ϵ U} ⊂Ω ⇒ p(U) ⊂q(U) . Applications of this result to differential inequalities, differential subordinations and integral inequalities are presented.

Univalence of Gaussian and confluent hypergeometric functions

Conditions are determined for the univalence convexity and star- likeness of Gaussian and confluent hypergeometric functions. In addition, sub- ordination results are obtained for these classes of functions.

Classes of univalent integral operators

Abstract Let A denote the set of functions f(z) = z + a2z2 + ··· that are analytic in the unit disc, and let S denote the subset of A consisting of univalent functions. With suitable conditions on the constants α, β, γ, and δ, and on the analytic functions φ(z) and Φ(z), the authors show that the integral operator I (f)(z) 7equiv; β + γ z γ; φ (z) ∫ 0 z f α (t) φ (t) t δ −1 dt 1 β = z + b 2 z 2 + … maps certain subsets of A into S. This result is then modified to obtain integral operators mapping S ∗ , K, S ∗ × K, and K × K into S ∗ . Here S ∗ and K denote the subsets of S consisting of starlike and convex functions, respectively.

Briot-bouquet differential equations and differential subordinations

Let β and γ be complex numbersn a positive integer, and let h be analytic in the unit disk U. This article describes conditions under which the Briot-Bouquet differential equation has analytic and univalent solutions. If h is also univalent then conditions are determined for the best dominants of the Briot-Bouquet differential subordination In addition, other differential subordinations are considered, together with applications to integral operators and to univalent functions.

Double integral starlike operators

The authors use the theory of differential subordinations to determine conditions on the kernel function W so that the function defined by is a starlike function.

Libera transform of functions with bounded turning

Conditions are determined for the starlikeness of the Libera transform of functions of bounded turning. In addition, several other differential subordinations and differential inequalities are considered.

The Hardy classes for functions in the class MV[α, k]

Abstract Suppose that f(z) = z + a2z2 + ··· + anzn + ··· is regular in the unit disc D with [f(z) f′ (z) z] ≠ 0 in D , and further let α ⩾ 0 and k ⩾ 2. If ∝ 2π o ¦ Re {(1 − α)z[ f′(z) f(z) ] + α(1 + z[ f″(z) f′(z) ])}¦ dθ ⪕ kπ for z ϵ D , then f(z) is said to belong to the class MV[α, k]. This class contains many of the special classes of regular and univalent functions. The authors determine the Hardy classes of which f(z), f′(z) and f″(z) belong and obtain growth estimates of an.