Constrained coefficients problem for generalized typically real functions
08 Apr 2016-Complex Variables and Elliptic Equations (Taylor & Francis)-Vol. 61, Iss: 9, pp 1303-1313
TL;DR: In this article, the main purpose of a paper is a solution of coefficients problems in. Problem related to the well-known Zalcman conjecture is presented, where the generalized Chebyshev polynomials of the second kind are defined as follows.
Abstract: For let denote the class of generalized typically real functions i.e. the class of functions of a formwhere , and is the unique probability measure on the interval . For the same range of parameters, let the generalized Chebyshev polynomials of the second kind be defined as followsWe see thatThe main purpose of a paper is a solution of coefficients problems in . Problem related to the well-known Zalcman conjecture is presented.
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01 Jan 2012
Abstract: Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties of obtained orthogonal polynomials.
7 citations
TL;DR: In this paper, the authors focus on determining properties of generalized Chebyshev polynomials of the first and second kind, and highlight some important results and connections between these two types.
Abstract: Our consideration is focused on determining properties of generalized Chebyshev polynomials of the first and second kind, sparking interest in constructing a theory similar to the classical one. This studies highlight some important results and connections between this two types. The paper is also concerned with the connection between orthogonal polynomials and typically real function, both strictly related to the Koebe function.
6 citations
TL;DR: In this paper, an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail is presented, and the behavior of functions related to generalized Pascal snails is studied.
Abstract: We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as the Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behaviour of functions related to generalized Pascal snail is studied.
4 citations
TL;DR: In this paper, the authors studied the family of normalized analytic error functions defined by Erf(z)=πz2erf (z)=z+∑n=2∞(− 1)n− 1(2n−1)(n − 1)!zn.
Abstract: Abstract In this present investigation, we will concern with the family of normalized analytic error function which is defined by Erf(z)=πz2erf(z)=z+∑n=2∞(−1)n−1(2n−1)(n−1)!zn. $$\begin{array}{} \displaystyle E_{r}f(z)=\frac{\sqrt{\pi z}}{2}\text{er} f(\sqrt{z})=z+\overset{\infty }{\underset {n=2}{\sum }}\frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}. \end{array}$$ By making the use of the trigonometric polynomials Un(p, q, eiθ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.
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Cites background from "Constrained coefficients problem fo..."
...The coefficient problem for generalized typically real functions provides one motivation to study properties of Chebyshev polynomials (see [6], [7], [13])....
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TL;DR: In this article, an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail is presented, and the behavior of functions related to generalized Pascal snails are demonstrated.
Abstract: We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behavior of functions related to generalized Pascal snail are demonstrated.
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TL;DR: For the class of normalized holomorphic and univalent functions f(z) in the unit disk D, this article showed that the conjecture holds for star-like functions and real functions with real coefficients.
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