For α ≥ 0 let $ℱ_α$ denote the class of functions defined for |z| < 1 by integrating $1/(1-xz)^α$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf'(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to $ℱ_0$. In general, it is only known that bounded starlike functions belong to $ℱ_α$ for α > 0.
In this paper, we define a function \(F : D\times D\times D\to \mathbb{C}\) in terms of \(f\) and show that Re\(F > 0\) for all \(\zeta,z,w \in D\) if and only if \(f\) belongs to the class of convex meromorphic functions.
Recently, Haji Mohd and Darus [1] revived the study of coefficient problems for univalent functions associated with quasi-subordination. Inspired largely by this article, we provide coefficient estimates with k-th root transform for certain subclasses of 𝒮 defined by quasi-subordination.
We study explicit examples of Loewner chains generated by absolutely continuous driving measures, and discuss how properties of driving measures are reflected in the shapes of the growing Loewner hulls.
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We consider the class 𝓩(k;w), k ∈ [0,2], w ∈ ℂ, of plane domains Ω called k-starlike with respect to the point w. An analytic characterization of regular and univalent functions f such that f(U) is in 𝓩(k;w), where w ∈ f(U), is presented. In particular, for k = 0 we obtain the well known analytic condition for a function f to be starlike w.r.t. w, i.e. to be regular and univalent in U and have f(U) starlike w.r.t. w ∈ f(U).
For functions of the form \[f(z) = z^{p} + \sum_{n = 1}^{\infty} a_{p + n} z^{p + n}\] we obtain sharp bounds for some coefficients functionals in certain subclasses of starlike functions. Certain applications of our main results are also given. In particular, Fekete-Szego-like inequality for classes of functions defined through extended fractional differintegrals are obtained.
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