THE IMPLEMENTATION OF DIFFERENTIAL SUBORDINATION AND SUPERORDINATION THEOREMS FOR ACHIEVING POSITIVE ANALYTIC FUNCTIONS

Authors:

D. Madhusudana Reddy,E. Keshava Reddy,

DOI NO:

https://doi.org/10.26782/jmcms.2020.03.00004

Keywords:

convex function,Star like function function,Differential subordination,Differential super ordination,

Abstract

Suppose  assume that complex numbers  such thatas  well as well as  Where The conditions are satisfactory for analytic function Cauchy’s Riemann equations are satisfied for the functions Here observe that the most excellent  subordinate and best leading. The applications are applied of those results are equivalent;ask as well as results to generalize and number of known results. By using a method based upon the Briot-Bouquet differential subordination, we prove several subordination results involving starlike and convex functions of complex order. Some special cases and consequences of the main subordination results are also indicated [I]. The main object of the present sequel to the aforementioned works is to apply a method based upon the Briot-Bouquet differential subordination in order to derive several subordination results involving starlike and convex functions of complex order[II],[III]. We also indicate some interesting special cases and consequences of our main subordination results.

Refference:

I. Ch. Pommerenke, Univalent Functions, Vanderhoeck and Ruprecht, GÄotingen, 1975.
II. H. M. Srivastava, A. A. Attiya, Some applications of di®erential subordination, Appl. Math. Letters, 20, 2007, 1142-1147.
III. J. L. Li, S. Owa, Sufficient conditions for starlikeness, Indian J. Pure appl.Math., 33(3), 2002, 313-318.
IV. K. S. Padmanabhan, On Su±cient conditions for starlikeness, Indian J.pure appl. Math., 32(4), 2001, 543-550.Di®erential subordination and superordination theorems . . . 157
V. L. Brickman, φ-like functions. I, Bull. Amer. Math.Soc., 79,1973, 555-558.
VI. M. S. Robertson, Certain classes of starlike functions, Michigan Math.J.,32, 1985, 135-140.
VII. M. Ali Rosihan , V. Ravichandran, M. Hussain Khan, K. G. Subramanian, Differential Sandwich Theorems for Certain Analytic Functions, Far East J. Math. Sci. 15(1), 2004, 87-94.
VIII. M. Obradovic, N. Tuneski, On the Starlike Criteria defined by Silverman, ZeszytyNauk. Politech. Rzeszowskiej. Mat., 181(24), 2000, 59-64.
IX. M. Obradovic, S. B. Joshi, I. Jovanovic, On Certain su±cient Conditions for Starlikeness and Convexity, Indian J. pure appl. Math., 29(3), 1998, 271-275.
X. N. E. Cho, J.Kim, On a sufficient Condition and as Angular Estimation for φ-likefunctions , Taiwan. J. Math., 2(4), 1998, 397-403.
XI. N. Tuneski, On the quotient of the representations of convexity and starlikeness, Math. Nachr., 248-249(1), 2003, 200-203.
XII. N. Tuneski, On a criteria for starlikeness of analytic functions, Integral Transforms and Special Functions, 14(3), 2003, 263-270.
XIII. N. Tuneski, On certain sufficient conditions for starlikeness, Internat.J.Math. & Math. Sci., 23(8), 2000, 521-527.
XIV. N. Tuneski, On Some Simple Sufficient Conditions for Univalence, MathematicaBohemica, 126(1), 2001, 229-236.
XV. St. Ruscheweyh, A subordination theorem for Á-like functions, J. London Math. Soc., 2(13), 1976, 275-280.
XVI. S. S. Miller, P.T. Mocanu, Differential Superordinations: Theory and Applications, Series on monographs and textbooks in pure and applied mathematics ( No. 225 ), Marcel Dekker, New York and Basel, 2000.
XVII. S. S. Miller, P. T. Mocanu, Di®erential subordination and Univalent functions, Michigan Math. J. 28, 1981, 157-171.
XVIII. T. N. Shanmugam, S. Sivasubramanian, M. Darus, Subordination and Superordination Results for Á-like Functions, J. Inequal. Pure and Appl. Math., 8, 2007, 1, 1-6.158 S. Singh, S. Gupta, S. Singh
XIX. T. Bulboaca, Classes of First-order Differential Superordinations, Demostratio Math., 35 (2), 2002, 287-292.
XX. T. Bulboaca, T.Tuneski, New Criteria for starlikeness and Strongly Starlikeness, Mathematica( Cluj
XXI. V. Ravichandran, Certain applications of ¯rst order di®erential subordination, Far East J. Math. Sci., 12(1), 2004, 41-51.
XXII. V. Ravichandran, N. Magesh, R. Rajalakshmi, On Certain Applications of Di®erential Subordinations for Á-like Functions, Tamkang J. Math.,36(2), 2005, 137-142.
XXIII. V. Ravichandran, C. Selvaraj, R. Rajalaksmi, Su±cient Conditions for Starlike Functions of Order ®, J. Inequal. Pure and Appl. Math. 3(5), 2002, 81, 1-6.
XXIV. V. Ravichandran, M. Darus, On a criteria for starlikeness, International Math. J., 4(2), 2003, 119-125.
XXV. V. Singh, N. Tuneski, On a Criteria for Starlikeness and Convexity of Analytic Functions, ActaMathematicaScientia, 24, 2004, 597-602.
XXVI. V. Singh, On some criteria for univalence and starlikeness, Indian J.Pure. Appl. Math. 34(4), 2003, 569-577.

View Download