CERTAIN DIFFERENTIAL SUPERORDINATIONS USING A GENERALIZED SĂLĂGEAN AND RUSCHEWEYH OPERATORS. Alb Lupaş Alina

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1 Acta Universitatis Apulensis ISSN: No. 25/211 pp CERTAIN DIFFERENTIAL SUPERORDINATIONS USING A GENERALIZED SĂLĂGEAN AND RUSCHEWEYH OPERATORS Alb Lupaş Alina Abstract. In the present paper we define a new operator using the generalied Sălăgean and Ruscheweyh operators. Denote by DR m the Hadamard product of the generalied Sălăgean operator D m and the Ruscheweyh operator Rm, given by DR m : A A, DRm f ) = Dm Rm ) f ) and A n = {f HU), f) = +a n+1 n , U} is the class of normalied analytic functions with A 1 = A. We study some differential superordinations regarding the operator DR m. 2 Mathematics Subject Classification: 3C45, 3A2, 34A4. Keywords: Differential superordination, convex function, best subordinant, differential operator. 1. Introduction and definitions Denote by U the unit disc of the complex plane U = { C : < 1} and HU) the space of holomorphic functions in U. Let A n = {f HU), f) = + a n+1 n , U} with A 1 = A and H[a, n] = {f HU), f) = a + a n n + a n+1 n , U} for a C and n N. If f and g are analytic functions in U, we say that f is superordinate to g, written g f, if there is a function w analytic in U, with w) =, w) < 1, for all U such that g) = fw)) for all U. If f is univalent, g f if and only if f) = g) and gu) fu). Let ψ : C 2 U C and h analytic in U. If p and ψ p ), p ) ; ) are univalent in U and satisfies the first-order) differential superordination h) ψp), p ); ), for U, 1) 31

2 A. Alb Lupaş - Certain differential superordinations using a generalied... p is called a solution of the differential superordination. The analytic function q is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q p for all p satisfying 1). An univalent subordinant q that satisfies q q for all subordinants q of 1) is said to be the best subordinant of 1). The best subordinant is unique up to a rotation of U. Definition 1 Al Oboudi [4]) For f A, and m N, the operator D m is defined by D m : A A, D f ) = f ) D 1 f ) = 1 ) f ) + f ) = D f ) D m... f) = 1 ) Dm 1 f ) + D m f )) = D D m 1 f ) ), for U. Remark 1 If f A and f) = + j=2 a j j, D m f ) = + j=2 [1 + j 1) ]m a j j, for U. Remark 2 For = 1 in the above definition we obtain the Sălăgean differential operator [7]. Definition 2 Ruscheweyh [6]) For f A, m N, the operator R m is defined by R m : A A, R f ) = f ) R 1 f ) = f )... m + 1) R m+1 f ) = R m f )) + mr m f ), U. Remark 3 If f A, f) = + j=2 a j j, R m f ) = + j=2 Cm m+j 1 a j j, U. Definition 3 [5]) We denote by Q the set of functions that are analytic and injective on U\E f), where E f) = {ζ U : limf ) = }, and f ζ) for ζ ζ U\E f). The subclass of Q for which f ) = a is denoted by Q a). We will use the following lemmas. Lemma 1 Miller and Mocanu [5]) Let h be a convex function with h) = a, and let γ C\{} be a complex number with Re γ. If p H[a, n] Q, p) + 1 γ p ) is univalent in U and h) p) + 1 γ p ), for U, q) p), for U, 32

3 A. Alb Lupaş - Certain differential superordinations using a generalied... γ n γ/n where q) = ht)tγ/n 1 dt, for U. The function q is convex and is the best subordinant. Lemma 2 Miller and Mocanu [5]) Let q be a convex function in U and let h) = q) + 1 γ q ), for U, where Re γ. If p H [a, n] Q, p) + 1 γ p ) is univalent in U and where q) = q) + 1 γ q ) p) + 1 γ p ), for U, q) p), for U, γ n γ/n ht)tγ/n 1 dt, for U. The function q is the best subordinant. 2. Main results Definition 4 [2]) Let and m N. Denote by DR m the operator given by the Hadamard product the convolution product) of the generalied Sălăgean operator D m and the Ruscheweyh operator Rm, DR m : A A, DR m f ) = Dm Rm ) f ). Remark 4 If f A and f) = + j=2 a j j, DR mf ) = + j=2 Cm m+j 1 [1 + j 1) ]m a 2 j j, for U. Remark 5 For = 1 we obtain the Hadamard product SR n [1] of the Sălăgean operator S n and Ruscheweyh operator R n. Theorem 1 Let h be a convex function, h) = 1. Let, m N, f A and suppose that m+1 m+1) DRm+1 f ) m1 ) m+1) DRm f ) is univalent and DRm f )) H [1, 1] Q. If h) m + 1 m 1 ) m + 1) DRm+1 f ) m + 1) DRm f ), for U, 2) q) DR m f )), for U, where q) = m+ 1 m+ 1 h t) tm 1+ 1 dt. The function q is convex and it is the best subordinant. Proof. With notation p ) = DR mf )) = 1+ j=2 Cm m+j 1 [1 + j 1) ]m ja 2 j j 1 and p ) = 1, we obtain for f) = + j=2 a j j, 33

4 A. Alb Lupaş - Certain differential superordinations using a generalied... p ) + p ) = m+1 m+1 p ) = m+1 and p ) + Evidently p H[1, 1]. Then 2) becomes DRm+1 f ) m m+1) DRm+1 h) p ) + ) DR m f )) m1 ) DR mf ) f ) m1 ) m+1) DRm f ). m + 1 p ), for U. By using Lemma 1 for γ = m + 1 and n = 1, we have q) p), for U, i.e. q) DR m f )), for U, where q) = m+ 1 m+ 1 h t) tm 1+ 1 dt. The function q is convex and it is the best subordinant. Corollary 1 [3]) Let h be a convex function, h) = 1. Let n N, f A and suppose that 1 SRn+1 f ) + n n+1 SRn f )) is univalent and SR n f )) H [1, 1] Q. If h) 1 SRn+1 f ) + n n + 1 SRn f )), for U, 3) q) SR n f )), for U, where q) = 1 Theorem 2 Let q be convex in U and let h be defined by h ) = q )+ m+1 q ),, m N. If f A, suppose that m+1 m+1) DRm+1 f ) m1 ) m+1) DR m f ) is univalent, DRm f )) H [1, 1] Q and satisfies the differential superordination h) = q )+ m + 1 q ) m + 1 m 1 ) m + 1) DRm+1 f ) m + 1) DRm f ), 4) for U, q) DR m f )), for U, where q) = m+ 1 m+ 1 h t) tm 1+ 1 dt. The function q is the best subordinant. Proof. Let p ) = DR mf )) = 1 + j=2 Cm m+j 1 [1 + j 1) ]m ja 2 j j 1. Differentiating, we obtain p ) + p ) = m+1 DRm+1 DR m f )) m1 ) DR n f ) and p ) + m+1 p ) = m+1 34 f ) m 1 + ) 1 f ) m+1) DRm+1

5 A. Alb Lupaş - Certain differential superordinations using a generalied... m1 ) m+1) DRm f ), for U and 4) becomes q) + m + 1 q ) p) + m + 1 p ), for U. Using Lemma 2 for γ = m + 1 and n = 1, we have q) p), for U, i.e. q) = m + 1 m+ 1 h t) t m 1+ 1 dt DR m f )), for U, and q is the best subordinant. Corollary 2 [3]) Let q be convex in U and let h be defined by h ) = q ) + q ). If n N, f A, suppose that 1 SRn+1 f ) + n n+1 SRn f )) is univalent, SR n f )) H [1, 1] Q and satisfies the differential superordination h) = q ) + q ) 1 SRn+1 f ) + q) SR n f )), for U, n n + 1 SRn f )), for U, 5) where q) = 1 ht)dt. The function q is the best subordinant. Theorem 3 Let h be a convex function, h) = 1. Let, m N, f A and suppose that DR mf )) is univalent and DRm f) H [1, 1] Q. If where q) = 1 Proof. h) DR m f )), for U, 6) q) DRm f ), for U, Consider p ) = DRm f) = + j=2 Cm m+j 1 [1+j 1)]m a 2 j j = 1 + j=2 Cm m+j 1 [1 + j 1) ]m a 2 j j 1. Evidently p H[1, 1]. We have p ) + p ) = DR m f )), for U. Then 6) becomes h) p) + p ), for U. By using Lemma 1 for γ = 1 and n = 1, we have q) p), for U, i.e. q) DRm f ), for U, 35

6 A. Alb Lupaş - Certain differential superordinations using a generalied... where q) = 1 Corollary 3 [3]) Let h be a convex function, h) = 1. Let n N, f A and suppose that SR n f )) is univalent and SRn f) H [1, 1] Q. If h) SR n f )), for U, 7) q) SRn f ), for U, where q) = 1 Theorem 4 Let q be convex in U and let h be defined by h ) = q ) + q ). If, m N, f A, suppose that DR mf )) is univalent, DRm f) H [1, 1] Q and satisfies the differential superordination where q) = 1 Proof. h) = q ) + q ) DR m f )), for U, 8) q) DRm f ), for U, ht)dt. The function q is the best subordinant. Let p ) = DRm f) = + j=2 Cm m+j 1 [1+j 1)]m a 2 j j = 1 + j=2 Cm m+j 1 [1 + j 1) ]m a 2 j j 1. Evidently p H[1, 1]. Differentiating, we obtain p) + p ) = DR m f )), for U and 8) becomes q) + q ) p) + p ), for U. Using Lemma 2 for γ = 1 and n = 1, we have q) p), for U, i.e. q) = 1 ht)dt DRm f ), for U, and q is the best subordinant. Corollary 4 [3]) Let q be convex in U and let h be defined by h ) = q )+q ). If n N, f A, suppose that SR n f )) is univalent, SRn f) H [1, 1] Q and satisfies the differential superordination h) = q ) + q ) SR n f )), for U, 9) q) SRn f ), for U, 36

7 A. Alb Lupaş - Certain differential superordinations using a generalied... where q) = 1 Theorem 5 ht)dt. The function q is the best subordinant. Let h) = 1+2β 1) 1+ be a convex function in U, where β < 1. Let, m N, f A and suppose that DR mf )) is univalent and DRm f) H [1, 1] Q. If h) DR m f )), for U, 1) q) DRm f ), for U, where q is given by q) = 2β β) ln1+), for U. The function q is convex and it is the best subordinant. Proof. Following the same steps as in the proof of Theorem and considering p) = DRm f), the differential superordination 1) becomes h) = 1 + 2β 1) 1 + p) + p ), for U. By using Lemma 1 for γ = 1 and n = 1, we have q) p), i.e., q) = 1 ht)dt = β 1)t 1 + t dt = 2β 1+21 β) 1 ln+1) DRm f ), for U. The function q is convex and it is the best subordinant. Theorem 6 Let h be a convex ) function, h) = 1. Let, m N, f A and DR suppose that m+1 f) is univalent and DRm+1 f) H [1, 1] Q. If DR m f) DR m f) where q) = 1 Proof. ) DR m+1 f ) h) DR mf ), for U, 11) q) DRm+1 f ), for U, f ) DR m Consider p ) = DRm+1 f) DR mf) = + j=2 Cm+1 m+j [1+j 1)]m+1 a 2 j j + = j=2 Cm m+j 1 [1+j 1)]m a 2 j j 1+ j=2 Cm+1 m+j [1+j 1)]m+1 a 2 j j 1 1+ j=2 Cm m+j 1 [1+j 1)]m a 2 j j 1. Evidently p H[1, 1]. 37

8 A. Alb Lupaş - Certain differential superordinations using a generalied... We have p ) = DRm+1 f)) DR mf) Then p ) + p ) = Then 11) becomes DR m+1 f) DR m f) ). p ) DRm f)) DR m f). h) p) + p ), for U. By using Lemma 1 for γ = 1 and n = 1, we have q) p), for U, i.e. q) DRm+1 f ), for U, f ) DR m where q) = 1 Corollary 5 [3]) Let h ) be a convex function, h) = 1. Let n N, f A and suppose that SR n+1 f) SR n f) is univalent and SR n+1 f) SR n f) H [1, 1] Q. If h) SR n+1 ) f ) SR n, for U, 12) f ) q) SRn+1 f ) SR n, for U, f ) where q) = 1 Theorem 7 Let q be convex in U and let h be defined ) by h ) = q ) + q ). DR If, m N, f A, suppose that m+1 f) is univalent, DRm+1 f) DR m f) H [1, 1] Q and satisfies the differential superordination where q) = 1 Proof. DR m f) ) DR h) = q ) + q m+1 f ) ) DR mf ), for U, 13) q) DRm+1 f ), for U, f ) DR m ht)dt. The function q is the best subordinant. Let p ) = DRm+1 f) DR mf) = + j=2 Cm+1 m+j [1+j 1)]m+1 a 2 j j + = j=2 Cm m+j 1 [1+j 1)]m a 2 j j 1+ j=2 Cm+1 m+j [1+j 1)]m+1 a 2 j j 1 1+ j=2 Cm m+j 1 [1+j 1)]m a 2 j j 1. Evidently p H[1, 1]. 38

9 A. Alb Lupaş - Certain differential superordinations using a generalied... Differentiating, we obtain p) + p ) = becomes DR m+1 f) q) + q ) p) + p ), for U. Using Lemma 2 for γ = 1 and n = 1, we have q) p), for U, i.e. q) = 1 DR m f) ), for U and 13) ht)dt DRm+1 f ), for U, f ) DR m and q is the best subordinant. Corollary 6 [3]) Let q be convex in U and ) let h be defined by h ) = q )+q ). If n N, f A, suppose that SR n+1 f) SR n f) is univalent, SRn+1 f) SR n f) H [1, 1] Q and satisfies the differential superordination h) = q ) + q ) SR n+1 ) f ) SR n, for U, 14) f ) q) SRn+1 f ) SR n, for U, f ) where q) = 1 ht)dt. The function q is the best subordinant. Theorem 8 Let h) = 1+2β 1) 1+ be a convex function in U, where β < 1. Let ) DR, m N, f A and suppose that m+1 f) is univalent, DRm+1 f) H [1, 1] Q. If DR m f) DR m f) ) DR m+1 f ) h) DR mf ), for U, 15) q) DRm+1 f ), for U, f ) DR m where q is given by q) = 2β β) ln1+), for U. The function q is convex and it is the best subordinant. Proof. Following the same steps as in the proof of Theorem and considering f) p) = DRm+1 f), the differential superordination 15) becomes DR m h) = 1 + 2β 1) 1 + p) + p ), for U. 39

10 A. Alb Lupaş - Certain differential superordinations using a generalied... By using Lemma 1 for γ = 1 and n = 1, we have q) p), i.e., q) = 1 ht)dt = β 1)t dt = 2β 1+21 β) t for U. The function q is convex and it is the best subordinant. References DRm+1 f ) ln+1) DR mf ), [1] Alina Alb Lupaş, Adriana Cătaş, Certain differential subordinations using Sălăgean and Ruscheweyh operators, Proceedings of International Conference on Complex Analysis and Related Topics, The 11th Romanian-Finish Seminar, Alba Iulia, 28, to appear). [2] Alina Alb Lupaş, Certain differential subordinations using a generalied Sălăgean operator and Ruscheweyh operator, submitted 29. [3] Alina Alb Lupaş, Certain differential superordinations using Sălăgean and Ruscheweyh operators, Analele Universităţii din Oradea, Fascicola Matematica, Tom XVII, Issue no. 2, 21, [4] F.M. Al-Oboudi, On univalent functions defined by a generalied Sălăgean operator, Ind. J. Math. Math. Sci., 24, no.25-28, [5] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 1, 23, [6] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., ), [7] G.St. Sălăgean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, ), Alb Lupaş Alina Department of Mathematics University of Oradea str. Universităţii nr. 1, 4187, Oradea, Romania dalb@uoradea.ro 4