Subordination by convex functions

Subordination by convex functions

Subordination by Convex Functions

Let K(α), 0≤α< 1, denote the class of functions g(z)= z+∑∞n=2anzn which are regular and univalently convex of order α in the unit disc U . Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U , f(0) = 0, and f(z)+zf ′(z) < g(z)+zg′(z) in U , then (i) f(z) < g(z) at least in |z| < r0, r0 = √ 5/3 = 0.745 . . . if f ∈ K; and (ii) f...

Differential Subordination for Meromorphic Multivalent Quasi-convex Functions

We introduce new classes of meromorphic multivalent quasi-convex functions and find some sufficient differential subordination theorems for such classes in punctured unit disk with applications in fractional calculus.

On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, w...

Applications of subordination theory to starlike functions

Let $p$ be an analytic function defined on the open unit disc $mathbb{D}$ with $p(0)=1.$ The conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{C}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp'(z)/p(z)$ is subordinate to $e^{z}$. Similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of Bernoulli $|w^{2}-1|=1$ ...