Petru T. Mocanu

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.

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.

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.

Some Properties of Starlike Functions with Respect to Symmetric-Conjugate Points

Let A be tile class of all analytic functions in the unit disk U such that f(0)=f′(0)−1=0. A function f∈A is called starlike with respect to 2n symmetric-conjugate points if Rezf′(z)/fn(z)>0 for z∈U, where fn(z)=12n∑k=0n−1[ω−kf(ωkz)

A sufficient condition for starlikeness of order α

We obtain a sufficient condition for starlikeness of order α, |f � (z)−λ(f (z)/z)+ λ − 1| <M = Mn(λ, α), where λ ∈ (0, 1), α ∈ (0, 1) and the function f( z)= z + an+1zn+1 + ··· is analytic in the unit disc U.

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.

The radius of -convexity for the class of starlike univalent functions, -real

We use a result due to Gutljanski; to obtain the radius of a-convexity for the class S* of starlike univalent functions for real a.

Three-cornered hat harmonic functions

Let be a harmonic function in the unit disc, where f and g are holomorphic functions and g′(z) = zf′(z). We obtain a simple sufficient condition on f for the harmonic function h to be univalent and its image to be a three-cornered hat domain.

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.