DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS

doi:10.1016/S0252-9602(13)60024-7

Abstract

Abstract:

This article presents some new results on the class SLM of functions that are analytic in the open unit disc U = {z : |z| < 1} satisfying the conditions that

f(0) = 0, f′(0) = 1, and α （1 +zf´´(z)/f′(z)）+ (1 − α)zf′(z)/f(z) ∈p(U)
for all z ∈ U, where α is a real number and
p(z) =1 + τ ²z²/1 − τ z − τ ²z² (z ∈ U).
The number τ = (1 − √5)/2 is such that τ ² = 1 + τ . The class SLM introduced by J. Dziok, R.K. Raina, and J. Sok´o l [3, Appl. Math. Comput. 218 (2011), 996-1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.

Key words: Univalent functions, starlike functions, subordination, Fibonacci numbers, tri-sectrix of Maclaurin, conchoid of de Sluze

CLC Number:

30C45

Jacek DZIOK|Ravinder Krishna RAINA|Janusz SOKóL. DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS[J].Acta mathematica scientia,Series B, 2013, 33(3): 609-620.

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References

[1] Brannan D A, Kirwan W E. On some classes of bounded univalent functions. J London Math Soc, 1969, 1(2): 431–443

[2] Dziok J, Raina R K, Sok´o l J. Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers. Comp Math Appl, 2011, 61(9): 2605–2613

[3] Dziok J, Raina R K, Sok´o l J. On -convex functions related to shell-like functions connected with Fibonacci numbers. Appl Math Comput, 2011, 218: 996–1002

[4] Goodman A W. Univalent Functions. Vol I. Tampa, Florida: Mariner Publishing Co, 1983

[5] Goodman A W. On uniformly convex functions. Ann Polon Math, 1991, 56: 87–92

[6] Janowski W. Extremal problems for a family of functions with positive real part and some related families. Ann Polon Math, 1970, 23: 159–177

[7] Janowski W. Some extremal problems for certain families of analytic functions. Ann Polon Math, 1973, 28: 297–326

[8] Jurasiska R, Stankiewicz J. Coefficients in some classes defined by subordination to multivalent majorants.
Proceedings of Conference on Complex Analysis (Bielsko-Bia la, 2001), Ann Polon Math, 2003, 80: 163–170

[9] Kanas S, Wi´sniowska A. Conic regions and k-uniform convexity II. Folia Scient Univ Tech Resoviensis, 1998, 170: 65–78

[10] Kanas S, Wi´sniowska A. Conic regions and k-uniform convexity. J Comput Appl Math, 1999, 105: 327–336

[11] Kanas S, Wi´sniowska A. Conic domains and starlike functions. Rev Roumaine Math Pures Appl, 2000, 45(3): 647–657

[12] Ma W, Minda D. Uniformly convex functions. Ann Polon Math, 1992, 57: 165–175

[13] Miller S S, Mocanu P T. Differential subordinations: theory and applications, Series of Monographs and Textbooks in Pure and Applied Mathematics. Vol 225. New York/Basel: Marcel Dekker Inc, 2000

[14] Mocanu P T. Une propri´et´e de convexit´e g´en´eralis´ee dans la th´eorie de la repr´esentation conforme. Math-
ematica (Cluj), 1969, 11(34): 127–133

[15] Nevanlinna R. Über die konforme Abbildund Sterngebieten. Oversikt av Finska-Vetenskaps Societen Forhandlingar, 1921, 63A(6): 48–403

[16] Paprocki E, Sok´o l J. The extremal problems in some subclass of strongly starlike functions. Folia Scient Univ Techn Resoviensis, 1996, 157: 89–94

[17] Robertson M S. Certain classes of starlike functions. Michigan Math J, 1954, 76(1): 755–758

[18] Rønning F. Uniformly convex functions and a corresponding class of starlike functions. Proc Amer Math Soc, 1993, 118: 189–196

[19] Sakaguchi K. A note on p-valent functions. J Math Soc Japan, 1962, 14: 312–321

[20] Sok´o l J. On some subclass of strongly starlike functions. Demonstratio Math, 1998, 31(1): 81–86

[21] Sok´o l J, Stankiewicz J. Radius of convexity of some subclasses of strongly starlike functions. Folia Scient Univ Tech Resoviensis, 1996, 147: 101–105

[22] Stankiewicz J. Quelques probl`emes extr`emaux dans les classes des fonctions -angulairement `etoil`ees. Ann
Univ Mariae Curie-Sk lodowska, Sect A, 1966/1971, 20: 59–75

[23] Study E. Konforme Abbildung Einfachzusammenhangender Bereiche. B C Teubner, Leipzig und Berlin, 1913

[24] Dziok J, Raina R K, Sokol J. On a Class of Starlike Functions Related to a Shell-Like Curve Connected with Fibonacci Numbers. Mathematical and Computer Modelling, 2013, 57: 1203–1211

DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS

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