Graphing Examples of Starlike and Convex Functions of order β (original) (raw)
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2010
Adriana Cătaş Faculty of Sciences, University of Oradea, Romania acatas@gmail.com Abstract By making use of a multiplier transformation, a subclass of p-valent functions in the open unit disc is introduced. The main results of the present paper provide various interesting properties of functions belonging to the new subclass. Some of these properties include, for example, several coefficient inequalities and distortion bounds for the function class which is considered here. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided.
Coefficient Inequalities for Certain Classes of Analytic and Univalent Functions
J. Ineq. Pure and Appl. Math
For functions f (z) which are starlike of order α, convex of order α, and λ-spirallike of order α in the open unit disk U, some interesting sufficient conditions involving coefficient inequalities for f (z) are discussed. Several (known or new) special cases and consequences of these coefficient inequalities are also considered.
Matematicki Vesnik
The subclasses S*(ot, 3) and C*(a, 3) of T , the class of analytic and univalent functions of the form 00 /(s) = z-Y, \ a I s have been considered. Sharp results n=2 n concerning coefficients, distortion of functions belonging to S*(ot, 3) and C*(a, 3) are determined along with a representation formula for the functions in £*(a, 3) • Furthermore, it is shown that the classes S*{a, 3) and C" t (a, 3) are closed under arithmetic mean and convex linear combinations.
On Starlike and Convex Functions of Complex Order with Fixed Second Coefficient
2015
Let Fp(b,M) denote the class of functions f(z) = z + ∑∞ k=2 akz k which are analytic in the open unit disc U = {z : |z| < 1} and satisfy the inequality ∣∣∣∣∣∣∣ b− 1 + zf ′ (z) f(z) b −M ∣∣∣∣∣∣∣ < M for b 6= 0, complex,M > 1 2 , |a2| = 2p, 0 ≤ p ≤ ( 1 +m 2 ) |b| , m = 1− 1 M and for all z ∈ U. Further f(z) is in the class Gp(b,M) if zf ′ (z) is in the class Fp(b,M). In the present paper, we obtain lower bounds for the classes introduced above and apply them to determine γ-spiral radiu for functions of the class Fp(b,M) and γ-convex radius for functions of the class Gp(b,M). 2010 Mathematics Subject Classification. 30C45.
On the radii of convexity and starlikeness for some special classes of analytic functions in a disk
Ukrainian Mathematical Journal, 1996
= n / 2, we obtain the subclass Qc~ of almost convex functions w = W(z), W (0) = 0, W'(0) = 1, defined by the condition Re [(1-z2)W'(z)] > c~. Note that Qc~ c Qo and the class Qo coincides with the class of functions convex along the direction of the imaginary axis [2]. Consider the class Uc~ of functions w= F(z), F(0)= 0, F'(0)= 1, regular in E and such that F(z)= zgt'(z), where W(z) ~ Qc~-Note that U a c U 0 and U 0 coincides with the class of typically real functions whose radius of starlikeness was found by Libera by the Robertson method [3, 4].
A Sub Class of K – Uniformly Starlike Functions with Negative Coefficients
In this present paper we introduce a subclass of analytic functions with negative coefficients. We study coefficient bounds, distortion properties, covering theorem, extreme points, radius of close to convexity, star likeness, and convexity and integral transformations for the functions in this class. The results of this paper generalize many earlier results in this direction. Mathematics subject classification: Primary 30C45
CONVEX AND STARLIKE UNIVALENT FUNCTIONS
1. Introduction. Let (S) denote the class of functions/(z) = z + 2? anzn which are regular and univalent in \z\ < 1 and which map \z\ < 1 onto domains D(f). Let (C), (S*), and (K) represent the subclasses of (S) where D(f) are respectively, close-to-convex, starlike with respect to the origin, and convex. It follows that (K)<=(S*)<=(C)<=(S). We will simply say "starlike" when we mean starlike with respect to the origin, and the statement "f(z) is convex" will mean that the domain D(f) is convex. The abbreviations "i.o.i." and "n.s.c." have the usual meanings. Let (F) denote the class of functions p(z) which are regular and satisfy p(0)=l, Rep(z)>0 for \z\ < 1. The following results, which we will use repeatedly, are well