We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.
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For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes $C_β(α)$ defined as follows: a function f regular in U = {z: |z| < 1} of the form $f(z) = z + ∑_{n=1}^{∞} a_n z^n$, z ∈ U, belongs to the class $C_β(α)$ if $Re{e^{iβ}(1 - α²z²)f'(z)} < 0$ for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in $C_β(α)$ are examined.
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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.
In this paper, we introduce some subclasses of meromorphic functions in the punctured unit disc. Several inclusion relationships and some other interesting properties of these classes are discussed.
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