CERTAIN DIFFERENTIAL SUBORDINATIONS USING A GENERALIZED SĂLĂGEAN OPERATOR AND RUSCHEWEYH OPERATOR Alina Alb Lupaş Dedicated to Prof. H.M. Srivastava for the 70th anniversary Abstract In the present paper we define a new operator using the generalied Sălăgean operator and the Ruscheweyh operator. Denote by DR n the Hadamard product of the generalied Sălăgean operator D n and of the Ruscheweyh operator R n, given by DR n : A A, DRn f ) = Dn Rn ) f ), where A n = {f HU) : f) = + a n+1 n+1 +..., U} is the class of normalied analytic functions in the unit disc, with A 1 := A. We study some differential subordinations regarding the operator DR n. MSC 2010 : 30C45, 30A20, 34A40 Key Words and Phrases: differential subordination, convex function, best dominant, differential operator, convolution product 1. Introduction Denote by U the unit disc of the complex plane, U = { C : < 1} and by HU) the space of holomorphic functions in U. c 2010, FCAA Diogenes Co. Bulgaria). All rights reserved.
356 A. Alb Lupaş Let A n = {f HU) : f) = + a n+1 n+1 +..., U}, for n N and A 1 := A. { } Denote by K = f A : Re f ) f ) + 1 > 0, U the class of the normalied convex functions in U. If f and g are analytic functions in U, we say that f is subordinate to g, written f g, if there is a function w analytic in U, with w0) = 0 and w) < 1 for all U, such that f) = gw)) for all U. If g is univalent, then f g if and only if f0) = g0) and fu) gu). Let ψ : C 3 U C and h be an univalent function in U. If p is analytic in U and satisfies the second-order) differential subordination ψp), p ), 2 p ); ) h), for U, 1) then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p q for all p satisfying 1). A dominant q that satisfies q q for all dominants q of 1) is said to be the best dominant of 1). The best dominant is unique up to a rotation of U. Definition 1.1. Al Oboudi [2], generalied the Sălăgean operator) For f A, U, 0 and n N, the operator D n : A A is defined by: D 0 f ) = f ) D 1 f ) = 1 ) f ) + f ) = D f ) D n... f) = 1 ) Dn 1 f ) + D n f )) = D D n 1 f ) ). Remark 1.2. If f A and f) = + a j j, then D n f ) = + [1 + j 1) ] n a j j, for U. Remark 1.3. For = 1 in the above definition we obtain the Sălăgean differential operator [5]. Definition 1.4. Ruscheweyh [4]) For f A and n N, the operator R n is defined by R n : A A, R 0 f ) = f ) R 1 f ) = f )... n + 1) R n+1 f ) = R n f )) + nr n f ), for U.
CERTAIN DIFFERENTIAL SUBORDINATIONS... 357 Remark 1.5. If f A and f) = + a j j, then R n f ) = + Cn+j 1a n j j, for U. Lemma 1.6. Miller and Mocanu [3]) Let g be a convex function in U and let h) = g) + nαg ), for U, where α > 0 and n is a positive integer. If p) = g0) + p n n + p n+1 n+1 +..., for U, is holomorphic in U and p) + αp ) h), for U, then p) g) 2. Main results Definition 2.1. Let 0 and n N. Denote by DR n : A A the operator given by the Hadamard product the convolution product) of the generalied Sălăgean operator D n and the Ruscheweyh operator Rn : DR n f ) = Dn Rn ) f ), for any U and each nonnegative integer n. Remark 2.2. If f A and f) = + a j j, then DR n f ) = + Cn+j 1 n [1 + j 1) ] n a 2 j j, for U. Remark 2.3. For = 1 we obtain the Hadamard product SR n see [1]) of the Sălăgean operator S n and the Ruscheweyh operator R n. Theorem 2.4. Let g be a convex function such that g 0) = 1 and let h be the function h ) = g ) + g ), for U. If 0, n N, f A and the differential subordination n + 1 DRn+1 f ) holds for U, then n 1 ) DR n f ) n 1 + 1 ) DR n f )) h ), 2) DR n f )) g ), for U, 3)
358 A. Alb Lupaş and P r o o f. With the notation p ) = DR n f )) = 1 + Cn+j 1 n [1 + j 1) ] n ja 2 j j 1 p 0) = 1, we obtain for f) = + a j j : p ) + p ) = 1 + Cn+j 1 n [1 + j 1) ] n j 2 a 2 j j 1 = n + 1 = n + 1 + C n+1 n+j [1 + j 1) ]n+1 a 2 j j + n 1 Cn+j 1 n [1 + j 1) ] n a 2 j j 1 n 1 + 1 ) j DRn+1 f ) Cn+j 1 n [1 + j 1) ] n a 2 j 1 n 1 ) j n 1 + 1 ) DR n f )) n 1 ) DR n f ). We have p ) + p ) h ), for U. By using Lemma 1.6 we obtain p ) g ), for U, i.e. DR n f )) g ), for U and this result is sharp. Corollary 2.5. see [1]) Let g be a convex function such that g 0) = 1 and let h be the function h ) = g ) + g ), for U. If n N, f A and the differential subordination 1 SRn+1 f ) + n n + 1 SRn f )) h ) for U, 4) holds, then SR n f )) g ) for U Theorem 2.6. Let g be a convex function, g 0) = 1 and let h be the function h ) = g ) + g ), for U. If n N and f A verifies the differential subordination DR n f )) h ) for U, 5) then DR nf ) g ) for U, 6)
CERTAIN DIFFERENTIAL SUBORDINATIONS... 359 P r o o f. For f A and f) = + a j j we have Consider DR n f ) = + Cn+j 1 n [1 + j 1) ] n a 2 j j, for U. p ) = DRn f ) = + Cn n+j 1 [1 + j 1) ]n a 2 j j = 1 + Cn+j 1 n [1 + j 1) ] n a 2 j j 1. We have p ) + p ) = DR n f )), for U. Then DR n f )) h ), for U, becomes p ) + p ) h ) = g ) + g ), for U. By using Lemma 1.6 we obtain p ) g ), for U, i.e. DRn f) g ), for U. Corollary 2.7. see [1]) Let g be a convex function, g 0) = 1 and let h be the function h ) = g ) + g ), for U. If n N and f A verifies the differential subordination then SRn f) SR n f )) h ), for U, 7) g ), for U, Theorem 2.8. Let g be a convex function such that g 0) = 1 and let h be the function h ) = g ) + g ), for U. If n N and f A verifies the differential subordination ) DR n+1 f ) DR nf ) h ), for U, 8) then DR n+1 f ) DR nf ) g ), for U, 9) P r o o f. For f A and f) = + a j j we have Consider DR n f ) = + Cn+j 1 n [1 + j 1) ] n a 2 j j, for U.
360 A. Alb Lupaş p ) = DRn+1 f ) DR nf ) = + Cn+1 n+j [1 + j 1) ]n+1 a 2 j j + Cn n+j 1 [1 + j 1) ]n a 2 j j = 1 + Cn+1 n+j [1 + j 1) ]n+1 a 2 j j 1 1 + Cn n+j 1 [1 + j 1) ]n a 2. j j 1 We have p ) = DRn+1 f)) DR nf) Then p ) + p ) = p ) DRn f)) DR n f). DR n+1 f) DR n f) ). Relation 8) becomes p ) + p ) h ) = g ) + g ), for U, and, by using Lemma 1.6 we f) obtain p ) g ), for U, i.e. DRn+1 DR nf) g ), for U. Corollary 2.9. see [1]) Let g be a convex function such that g 0) = 1 and let h be the function h ) = g ) + g ), for U. If n N and f A verifies the differential subordination SR n+1 ) f ) SR n h ) for U, 10) f ) then SRn+1 f) SR n f) g ), for U, References [1] A. Alb Lupaş, Certain differential subordinations using Sălăgean and Ruscheweyh operators. Mathematica Cluj) Proc. Intern. Conference on Complex Analysis and Related Topics, The 11th Romanian-Finnish Seminar, Alba Iulia, 2008). To appear. [2] F.M. Al-Oboudi, On univalent functions defined by a generalied Sălăgean operator. Ind. J. Math. Math. Sci. 2004, No 25-28 2004), 1429-1436. [3] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications. Marcel Dekker Inc., New York, Basel, 2000. [4] St. Ruscheweyh, New criteria for univalent functions. Proc. Amer. Math. Soc. 49 1975), 109-115. [5] G. St. Sălăgean, Subclasses of univalent functions. Lecture Notes in Math., 1013 1983), 362-372 Springer Verlag, Berlin). Department of Mathematics, University of Oradea Str. Universităţii, No.1 410087 Oradea ROMANIA e-mail: dalb@uoradea.ro Received: GFTA, August 27-31, 2010