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Let A denote the class of normalized analytic functions in the unit disc U = z: |z| < 1. The author obtains fixed values of δ and ϱ (δ ≈ 0.308390864..., ϱ ≈ 0.0903572...) such that the integral transforms F and G defined by $F\left(z\right)={\int }_0^z(f\left(t\right)/t)dt$ and $G\left(z\right)=(2/z){\int }_0^{z}g\left(t\right)dt$ are starlike (univalent) in U, whenever f ∈ A and g ∈ A satisfy Ref’(z) > -δ and Re g’(z) > -ϱ respectively in U.

Let denote the open unit disk and f: → ℂ̅ be meromorphic and univalent in with a simple pole at p ∈ (0,1) and satisfying the standard normalization f(0) = f’(0)-1 = 0. Also, assume that f has the expansion $f\left(z\right)={\sum }_{n=-1}^{\infty }aₙ(z-p)ⁿ$, |z-p| < 1-p, and maps onto a domain whose complement with respect to ℂ̅ is a convex set (starlike set with respect to a point w₀ ∈ ℂ, w₀ ≠ 0 resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by $Co\left(p\right)({\Sigma }^s(p,w₀)$ resp.). We prove some coefficient...

For n ≥ 1, let denote the class of all analytic functions f in the unit disk Δ of the form $f\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$. For Re α < 2 and γ > 0 given, let (γ,α) denote the class of all functions f ∈ satisfying the condition |f’(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in (γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.

Let 𝓐 denote the class of all normalized analytic functions f (f(0) = 0 = f'(0)-1) in the open unit disc Δ. For 0 < λ ≤ 1, define 𝓤(λ) = {f ∈ 𝓐 : |(z/f(z))²f'(z) - 1| < λ, z ∈ Δ} and 𝓟(2λ) = f ∈ 𝓐 : |(z/f(z))''| < 2λ, z ∈ Δ.cr Recently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding the order...

For μ ∈ ℂ such that Re μ > 0 let ${ℱ}_{\mu }$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $Re(2\pi /\mu z{f}^{\text{'}}\left(z\right)/f\left(z\right)+(1+z)/(1-z))>0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class ${ℱ}_{\mu }\left(\lambda \right)=f\in {ℱ}_{\mu }:f^{\text{'}}\left(0\right)=(\mu /\pi )(\lambda -1)and{f}^{\text{'}\text{'}}\left(0\right)=(\mu /\pi )\left(a\right(1-\left|\lambda \right|²)+(\mu /\pi )(\lambda -1)²-(1-\lambda ²))$. In the final section we graphically illustrate the region of variability for several sets of parameters.

Let represent the class of all normalized analytic functions f in the unit disc Δ. In the present work, we first obtain a necessary condition for convex functions in Δ. Conditions are established for a certain combination of functions to be starlike or convex in Δ. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and μ so that the non-linear integral transform...

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings f from...

In a recent paper Fournier and Ruscheweyh established a theorem related to a certain functional. We extend their result differently, and then use it to obtain a precise upper bound on α so that for f analytic in |z| < 1, f(0) = f'(0) - 1 = 0 and satisfying Re{zf''(z)} > -λ, the function f is starlike.

Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f’(0) − 1 = 0. For β < 1, let $P{⁰}_{\beta }=f\in A:Re{f}^{\text{'}}\left(z\right)>\beta ,z\in \Delta$. For λ > 0, suppose that denotes any one of the following classes of functions: $M_{1,\lambda }^{\left(1\right)}=f\in :Rez{\left(z{f}^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(2\right)}=f\in :Rez{\left(z²f^{\text{'}\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(3\right)}=f\in :Re1/2{\left(z{\left(z²f^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}\right)}^{\text{'}}-1>-\lambda ,z\in \Delta$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in ${}_{\gamma }$ or ${}_{\gamma }$, γ ∈ [0,1/2]. Here ${}_{\gamma }$ and ${}_{\gamma }$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain a number...