On a subclass of strongly starlike functions (original) (raw)
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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].
On the radius of α-convexity of certain classes of starlike functions
1977
Let M (a) denote the class of a-convex functions, a real, that is the class of analytic functions f (z) = z + E2 a" z" in the unit disc .D = {z: z < 1} which satisfies in D the condition f' (z) f (z)/z 0 and Re {(1_a) ' f'(z) + a 1 + ()} z f, (z) > 0. Let W (a) .f (z) ff z) denote the class of meromo-phic a-convex functions. a real, that is the class of analytic functions (z) = z + E, b" z' in D' _ {z: 0 < z < 1} which satisfies in D* the conditions zrAz)/^'.(z)-0 and Re ((l-a) (z^) + a f 1 +-z1` ;(z z) ^ G 0. In this paper we obtain the relation bctwcen M (a) and W (a). The radius of a-convexity for certain classes of starlike functions is also obtained.
Radius of Starlikeness of Certain Class of Close-To-Convex Functions
International Journal of Pure and Apllied Mathematics, 2017
Let G(α, δ) be the class of normalized analytic functions by f (0) = f ′ (0) − 1 = 0 and defined in the open unit disc, E = {z : |z| < 1} satisfying Re{e iα 2zf ′ (z) f (z) − f (−z) } > δ, for |α| < π , cos α − δ > 0 and 0 ≤ δ < 1. In this paper, we have determined the bound for f (z) z and have obtained result for radius of starlikeness, RSt of G(α, δ).
Some conditions on starlike and close to convex functions
Thermal Science
Many mathematical concepts are explained when viewed through complex function theory. We are here basically concerned with the form f(z)=a0+a1z+a2z2+...f(z) ?A, f(z)=z+??, n=2anZ2 will be an analytic function in the open unit disc U={z:|z|<1, z=?C}normalized by f(0) = 0, f'(0)=1. In this work, starlike functions and close-to-convex functions with order 1/4 have been studied according to the exact analytic requirements.
2012
Several radii problems are considered for functions f (z) = z + a 2 z 2 + · · · with fixed second coeffcient a 2 . For 0 ≤ β < 1, sharp radius of starlikeness of order β for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order β for uniformly convex functions, and sharp radius of strong-starlikeness of order γ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.
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.
Starlikeness criteria for a certain class of analytic functions
Applied Mathematics Letters, 2011
We denote by A, the class of all analytic functions f in the unit disc ∆ = {z ∈ C : |z| < 1} with the normalization f (0) = f ′ (0) − 1 = 0. For a positive number λ > 0, we denote by U 3 (λ) the class of all f (z) = z + ∑ ∞ n=2 a n z n ∈ A, such that a 3 − a 2 2 = 0, and satisfying the condition z f (z) 2 f ′ (z) − 1 < λ, z ∈ ∆. A function f ∈ A is said to be in SR(γ) if | arg f ′ (z)| < π γ /2. In this paper, we find conditions on λ, α and γ such that U 3 (λ) is included in the class of all starlike functions of order α, or the class of all strongly starlike functions of order γ , or SR(γ), respectively.
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.
Certain results of starlike and convex functions in some conditions
Thermal Science
The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+??, n?(anzn) converges for all complex numbers z with |z|<1 and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)=f'(0)=1, and g(0)=0, g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain D?C never gets the same value twice; that is, if f(z1)-f(z2)?0 for all z1 and z2 with z1 ? z2. Let A be the class of analytic functions in the unit disk U={z:|z|<1} that are normalized with f(0)=F'(0)=1. In this paper we give the some necessary conditions for f(z) ? S* [a, a2] and 0?a2?a?1 f'(z)(2r-1)[1-f'(z)]+zf?(z / 2r[f'(z)]2. This condition means that convexity and ...