denote the subclass of the functions f H of the form H of the form

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1 Scholars Journal of Engineering and Technology (SJET) Sch. J. Eng. Tech., 05; 3(4C):55-59 Scholars Academic and Scientific Publisher (An International Publisher for Academic and Scientific Resources) ISSN 3-435X (Online) ISSN (Print) Research Article On sandwich theorems for certain analytic functions Jing Wang, Lifeng Guo School of Mathematica Science and Technology, Northeast Petroleum University, Daqing 6338, China. *Corresponding author Jing Wang Abstract: In this paper, we derive some subordination and superordination results for certain normalied analytic functions in the open unit disk. Keywords: univalent functions; starlike functions; subordination; superordination INTRODUCTION Let H denote the class of analytic functions in the open unit disc U { C }, and H[ a n] denote the subclass of the functions f H of the form Also,Let A be the subclass the functions f f () a a a n n n n () H of the form f () a a () 3 3 A function f A is said to be in the class S of starlike functions in U, if it satisfies the inequality Re( ) 0 U Furthermore, a function f A is said to be in the class C of convex functions in U, if it f ( ) f ( ) satisfies the inequality Re( ) 0 U f ( ) f( ) Let and F () be analytic in U, we say that the function is subordinate to F () in U, if there exists an analytic function w () in U such that w(), and f ( ) F( w( )), denoted by f p F or f ( ) p F( ). If F () is univalent in U, the subordination is equivalent to f(0) F(0) and f ( U) F( U). 3 Let ph H and let ( rst ) C U C. If p and ( p( ) p( ) p ( ) ) are univalent and if p satisfies the second-order superordination h ( ) ( p( ) p ( ) p ( ) ) p (3) p is a solution of the differential superordination (3). (If f is subordinate to F, F is superordinate to f.) An analytic function q is called a subordinant if qp p for all p satisfying (3). A univalent subordinant Q that satisfies qp Q for all subordinants q of (3) is said to be the best subordinant. Recently Miller and Mocanu [] obtained conditions on hq and for which the following implication holds: h( ) p ( p( ) p( ) p ( ) ) q( ) p p( ) (4) Using the results of Miller and Mocanu [], Bulboacǎ [] considered certain classes of first-order differential superordinations as well as superordination-preserving integral operators [3]. Ali et al. [4] have used the results of Bulboacǎ [] and obtained sufficient conditions for certain normalied analytic functions to satisfy 55

2 Jing Wang et al., Sch. J. Eng. Tech., 05; 3(4C):55-59 f () q( ) p p q( ) (5) where q and q are given univalent functions in U with q (0) and q (0). Shanmugam et al. [5] obtained sufficient conditions for normalied analytic functions to satisfy f ( ) f ( ) q ( ) p p q( ) andq ( ) p p q ( ) (6) f ( ) f ( ) where q and q are given univalent functions in U with q (0) and q (0), while Obradović and Owa [6] obtained subordination results with the quantity ( f ( ) ) (see also [7]). For 0, a function f ( ) N( ) if and only if f () A and f () Re{ ( ) } 0 U (7) f ( ) f ( ) N( ) was introduced by M.Obradović [8] recently, and he called this class of functions to be non-bailevič type. Tuneski and Darus [9] obtained Fekete-Segö inequality for the non-baileviˇc class of functions. Using this non- Bailevič class, Wang et al. [0] studied many subordination results for the class N( A B) defined as f ( ) A N( A B) { f ( ) A ( )( ) ( ) p U} (8) f ( ) f ( ) f ( ) B where 0 C B A B A R. The main object of the present sequel to the aforementioned works is to apply a method based on the differential subordination in order to derive several subordination results. Furthermore, we obtain the previous results of Srivastava and Lashin [7], Singh [], Shanmugam et al. [] and Obradović andowa [6] as special cases of some of the results presented here. Some lemmas To prove our main result, we will need the following definition and lemmas: DEFINITION [] Denote by the set of all functions that are analytic and injective on U E( f ), where E( f ) { U lim f ( ) } (9) and are such that f ( ) 0 for U E( f ). U and [5] Let q be univalent in U and let C with p( ) p q( ) and q is the best dominant. Re( ) max{0 } If p () is analytic in q( ) q( ) Re p( ) p( ) p q( ) q( ) (0) [] Let q be convex univalent in U and let C with Re( ) 0 If p( ) H[ q(0) ] p( ) p( ) is univalent in U, and q( ) q( ) p p( ) p( ) () q( ) p p( ) and q is the best subordinant. and 56

3 Jing Wang et al., Sch. J. Eng. Tech., 05; 3(4C):55-59 [3] Let q be univalent in U and let be analytic in a domain containing qu ( ) with ( w) 0 when w q( U). Set h( ) q( ) ( q( )), F( ) ( q( )) h( ) Suppose that () h () is starlike univalent in U ; () If Re( ) 0 for U F( ) h ( ) p( ) q( ) ( p( )) p( ) ( F( )) ( q( )) q( ) ( q( )) p and q() is the best dominant. p () [3] Let q be convex univalent in U, and let be analytic in a domain containing ( ) () q ( ) ( q( )) is starlike univalent in U ; ( q( ) () Re( ( q( )) ) 0 for U. If p( ) H[ q(0) ], with pu ( ) and ( p( )) p( ) ( p( )) is univalent in U and ( q( )) q( ) ( q( )) ( p( )) p( ) ( p( )) q( ) p p( ) and q is the best subordinant. Sandwich theroems By using., we first prove the following Theorem. qu. Suppose that p (3) Theorem 3..Let q () be univalent in U, 0 and C. Further assume that q() Re{ } max{0 Re } q() If f ( ) A g( ) S, g( ) g( ) f ( ) g( ) ( ) ( )( ) q( ) q( ) f ( ) g( ) f ( ) f ( ) implies that and q () is the best dominant. (4) p (5) g () ( ) p q ( ) (6) Proof. Define the function p () by g () p ( ) ( ) (7) Then a computation shows that g( ) g( ) f ( ) g( ) ( ) ( )( ) p( ) p( ) (8) f ( ) g( ) f ( ) f ( ) Then we obtain that p( ) p( ) p q( ) q( ) (9) By using.,we have the result. 57

4 Jing Wang et al., Sch. J. Eng. Tech., 05; 3(4C):55-59 Theorem 3..Let q () be convex univalent in U, 0, C ( ) Hq [ (0) ] and g( ) f ( ) be univalent in U. If and q () is the best subordinant. Proof. Let g( ) f ( ) with Re( ) 0. Suppose g( ) g( ) f ( ) g( ) ( ) ( )( ) (0) f ( ) g( ) f ( ) f ( ) g( ) g( ) f ( ) g( ) q( ) q( ) p ( ) ( )( ) () f ( ) g( ) f ( ) f ( ) g () q ( ) p ( ) () p ( ) ( ). Then Theorem 3. follows by an application of.. Combining the results of differential subordination and superordination, we obtain the following sandwich result. Corollary 3.3.Let () q be univalent and let q () be convex univalent in U, 0 and C with Re( ) 0. Suppose q () satisfies (4). If g( ) ( ) Hq [ (0) ], f ( ) ( ) ( )( ) is g ( ) g ( ) f ( ) g ( ) f ( ) g( ) f ( ) f ( ) univalent in U, and g( ) g( ) f ( ) g( ) q ( ) q ( ) p ( ) ( )( ) p q( ) q( ) (3) f ( ) g( ) f ( ) f ( ) g () q( ) p ( ) p q( ) (4) and q () and q () are respectively the best subordinant and the best dominant. For and g(), we get the following corollary. Corollary 3.4.Let () q be univalent and let q () be convex univalent in U, 0. Suppose q () g( ) (4). If ( ) f ( ) Hq [ (0) ], ( )( ) is univalent in U, and and () f ( ) q and q () f ( ) f ( ) satisfies f ( ) q ( ) q ( ) p ( )( ) p q( ) q( ) (5) f ( ) f ( ) q( ) p ( ) p q( ) (6) are respectively the best subordinant and best dominant. 4. Open Problem Let H be the class of analytic functions in U { C }, and H[ a p] be the subclass of H consisting of functions of the form p p f () a a a (7) p p Let Ap ( ) be the subclass of H consisting of functions of the form f () a a (8) p p p p p 58

5 Jing Wang et al., Sch. J. Eng. Tech., 05; 3(4C):55-59 A function f A( p) is said to be in the class S ( p) of p -valent starlike functions in U, if it satisfies the inequality Re( ) 0 U f ( ) pf ( ) Let f ( ) A( p) and g( ) S ( p) following implication holds:. We can consider sufficient conditions on hq q and for which the g () q( ) p ( ) p q( ) (9) or ( ) f ( ) f ( ) q( ) p ( ) p q( ) (30) g () where 0 and 0. REFERENCES. Miller SS, Mocanu PT; Subordinants of differential superordinations. Complex variables, 003; 48(0): Bulboacǎ T; Classes of first-order differential superordinations, Demonstratio Mathematica, 00; 35(): Bulboacǎ T; A class of superordination-preserving integral operators, IndagationesMathematicae. New Series, 00; 3(3): M Ali R, Ravichandran V, Hussain Khan M, Subramanian KG; Differential Sandwich Theorems For Certain Analytic Functions. Far East Journal of Mathematical Sciences, 004; 5(): Shanmugam TN, Ravichandran V, Sivasubramanian S; Differential sandwich theorems for some subclasses of analytic functions, The Australian Journal of Mathematical Analysis and Applications, 006; 3(). 6. Obradović M, Owa S; On certain properties for some classes of starlike functions, Journal of Mathematical Analysis and Applications, 990; 45(): Srivastava HM, Lashin AY; Some applications of the Briot-Bouquet differential subordination. J. Inequal. Pure Appl. Math, 005; 6(): Obradovic M, A class of univalent functions.hokkaido Math, 998; 7: Tuneski N, Darus M; Fekete-Segö functional for non-bailevič functions, ActaMathematica. Academiae Paedagogicae Nyíregyháiensis. New Series, 00; 8(): Wang Z, Gao C, Liao M; On certain generalied class of non-bailevič functions, ActaMathematica. Academiae Paedagogicae Nyíregyháiensis. New Series, 005; (): Singh V; On some criteria for univalence and starlikeness, Indian Journal of Pure and Applied Mathematics, 003; 34(4): Shanmugam TN, Sivasubramanian S, Silverman H; On sandwich theorems for some classes of analytic functions, International Journal of Mathematics and Mathematical Sciences, Miller SS, Mocanu PT; Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 000; 5. 59