Bull. Korean Math. Soc. 46 (2009), No. 5,. 905 915 DOI 10.4134/BKMS.2009.46.5.905 ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS INVOLVING A GENERALIZED DIFFERENTIAL OPERATOR Chellian Selvaraj and Kuathai A. Selvaumaran Abstract. Maing use of a generalized differential oerator we introduce some new subclasses of multivalent analytic functions in the oen unit dis and investigate their inclusion relationshis. Some integral reserving roerties of these subclasses are also discussed. 1. Introduction and reliminaries Let A denote the class of functions f(z) of the form (1) f(z) = z + a n z +n, ( N = {1, 2, 3,...}), which are analytic and -valent in the oen unit dis U = {z : z C and z < 1}. For functions f given by (1) and g given by g(z) = z + b n z +n, the Hadamard roduct (or convolution) of f and g is defined by (f g)(z) = z + a n b n z +n. Given two functions f and g, which are analytic in U, the function f is said to be subordinate to g in U if there exists a function w analytic in U with such that w(0) = 0, w(z) < 1 (z U), f(z) = g(w(z)) Received July 23, 2008; Revised January 21, 2009. 2000 Mathematics Subject Classification. 30C45, 30C50, 30C75. Key words and hrases. multivalent function, convex univalent function, starlie with resect to symmetric oints, Hadamard roduct, subordination, integral oerator. 905 c 2009 The Korean Mathematical Society
906 C. SELVARAJ AND K. A. SELVAKUMARAN We denote this subordination by f(z) g(z). Furthermore, if the function g is univalent in U, then f(z) g(z) (z U) f(0) = g(0) and f(u) g(u). Let P denote the class of analytic functions h(z) with h(0) = 1, which are convex and univalent in U and for which R{h(z)} > 0 Analogous to the oerator defined recently by Selvaraj and Santhosh Moni [6], we define an oerator D δ,g f on A as follows: For a fixed function g A given by (2) g(z) = z + b n z +n, (b n 0; N = {1, 2, 3,...}), D δ,g f(z) : A A is defined by (3) D 0,gf(z) = (f g)(z), If f(z) A, then we have D 1,gf(z) = (1 )(f g)(z) + z((f g)(z)), D δ,gf(z) = D 1,g(D δ 1,g f(z)). (4) D δ,gf(z) = z + ( 1 + n ) δ a n b n z +n, where δ N 0 = N {0} and 0. It easily follows from (3) that (5) z (Dδ,gf(z)) = D δ+1,g f(z) (1 )Dδ,gf(z). Throughout this aer, we assume that, N, ε = ex( 2πi ), and (6) f δ, = 1 1 ε j (D,gf(ε δ j z)) = z +, (f A ). Clearly, for = 1, we have f δ,1 = D δ,gf(z). Maing use of the oerator D,g δ f(z), we now introduce and study the following subclasses of A of -valent analytic functions. Definition. A function f A is said to be in the class S, δ (; g; h), if it satisfies (7) z(d,g δ f(z)) f, δ h(z) (z U), where h P and f, δ 0
ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS 907 Remar 1.1. If we let δ = 0 and g(z) = z lf m (α 1,..., α l ; β 1,..., β m ; z), then S, δ (; g; h) reduces to the function class Sl,m, (α 1; h) introduced and investigated by Zhi-Gang, Wang Yue-Ping Jiang, and H. M. Srivastava [10]. Remar 1.2. If we let δ = 0 and g(z) = z + (a) n (c) n z +n, then S δ, (; g; h) reduces to the function class T,(a, c; h) introduced and investigated by N-Eng Xu and Ding-Gong Yang [7]. Remar 1.3. Let g(z) = h(z) = 1 + z 1 z. Then S 0 1,2(; g; h) = S s. The class S s of functions starlie with resect to symmetric oints has been studied by several authors (see [3], [5], [9]). Definition. A function f A is said to be in the class K, δ (; g; h), if it satisfies (8) z(d,g δ f(z)) ϕ δ h(z) (z U), for some ϕ(z) S, δ (; g; h), where h P and ϕδ, 0 is defined as in (6). Definition. A function f A is said to be in the class C, δ (α, ; g; h), if it satisfies (9) (1 α) z(dδ,g f(z)) ϕ δ, + α (z(dδ,g f(z)) ) h(z) (z U) ) (ϕ δ, for some α (α 0) and ϕ(z) S δ, (; g; h), where h P and (ϕδ, ) 0. We need the following lemmas to derive our results. Lemma 1.4 ([1]). Let β (β 0) and γ be comlex numbers and let h(z) be analytic and convex univalent in U with R{βh(z) + γ} > 0 If q(z) is analytic in U with q(0) = h(0), then the subordination imlies that q(z) + zq (z) βq(z) + γ q(z) h(z) h(z) (z U)
908 C. SELVARAJ AND K. A. SELVAKUMARAN Lemma 1.5 ([2]). Let h(z) be analytic and convex univalent in U and let w(z) be analytic in U with R{w(z)} 0 If q(z) is analytic in U with q(0) = h(0), then the subordination q(z) + w(z)zq (z) h(z) (z U) imlies that q(z) h(z) Lemma 1.6. Let f(z) S, δ (; g; h). Then (10) z(f, δ ) f, δ h(z) Proof. For f(z) A, we have from (6) that f δ,(; g; ε j z) = 1 = εj 1 m=0 1 m=0 ε m D,gf(ε δ m+j z) ε (m+j) D,gf(ε δ m+j z) = ε j f δ,, (j {0, 1,..., 1}) and Hence (11) (f δ,) = 1 z(f, δ ) f, δ = 1 = 1 1 ε j(1 ) (D,gf(ε δ j z)). 1 1 ε j(1 ) z(d,g δ f(εj z)) f, δ ε j z(dδ,g f(εj z)) f δ, (; g; εj z) Since f(z) S, δ (; g; h), we have (12) ε j z(dδ,g f(εj z)) f δ, (; g; εj z) h(z) for j {0, 1,..., 1}. Noting that h(z) is convex univalent in U, from(11) and (12) we conclude that (10) holds true.
ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS 909 Theorem 2.1. Let h(z) P with 2. A set of inclusion relationshis (13) R{h(z)} > 1 1 (z U; > 1). If f(z) S δ+1, (; g; h), then f(z) Sδ, (; g; h) rovided f, δ 0 Proof. By using (5) and (6), we have (14) (1 )f δ,+ z (f δ,) = 1 Let f(z) S δ+1, (; g; h) and 1 (15) w(z) = z(f, δ ) f, δ. ε j (D δ+1,g f(εj δ+1 z)) = f,. Then w(z) is analytic in U, with w(0) = 1, and from (14) and (15) we have (16) 1 + w(z) = f δ+1, f, δ. Differentiating (16) with resect to z and using (15), we get (17) w(z) + zw (z) z(f = (1 ) + w(z) From (17) and Lemma 1.6 we note that (18) w(z) + δ+1, ). f δ+1, zw (z) h(z) (1 ) + w(z) In view of (13) and (18), we deduce from Lemma 1.4 that (19) w(z) h(z) Suose that q(z) = z(dδ,g f(z)) f, δ. Then q(z) is analytic in U, with q(0) = 1, and we obtain from (5) that (20) f δ,q(z) = 1 Dδ+1,g f(z) + (1 1 ) D δ,gf(z). Differentiating both sides of (20) with resect to z, we get ( (21) zq (z) + ( 1 1) + z(f, δ ) )q(z) f, δ = z(dδ+1,g f(z)) f, δ.
910 C. SELVARAJ AND K. A. SELVAKUMARAN Now, we find from (14), (15) and (21) that (22) q(z) + zq (z) z(d = (1 ) + w(z) f δ+1, δ+1,g f(z)) h(z) since f(z) S δ+1, (; g; h). From (13) and (19) we observe that { } R (1 ) + w(z) > 0. Therefore, from (22) and Lemma 1.5 we conclude that which shows that f(z) S, δ (; g; h). q(z) h(z) (z U) (z U), Theorem 2.2. Let h(z) P with (23) R{h(z)} > 1 1 (z U; > 1). If f(z) K δ+1, (; g; h) with resect to ϕ(z) Sδ+1, (; g; h), then f(z) K, δ (; g; h) rovided ϕδ, 0 Proof. Let f(z) K δ+1, (; g; h). Then there exists a function ϕ(z) Sδ+1, (; g; h) such that (24) z(d δ+1,g f(z)) ϕ δ+1, h(z) An alication of Theorem 2.1 yields ϕ(z) S, δ (; g; h) and Lemma 1.6 leads to (25) ψ(z) = z(ϕδ, ) ϕ δ, h(z) Let q(z) = z(dδ,g f(z)) ϕ δ,. By using (5), q(z) can be written as follows (26) ϕ δ,q(z) = 1 Dδ+1,g (1 f(z) + 1 ) D δ,gf(z). Differentiating both sides of (26) with resect to z and using (14) (with f relaced by ϕ), we get (27) q(z) + zq (z) z(d = (1 ) + ψ(z) ϕ δ+1, δ+1,g f(z)).
ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS 911 Now, from (24) and (27) we find that (28) q(z) + zq (z) h(z) (1 ) + ψ(z) Combining (23), (25) and (28), we deduce from Lemma 1.5 that q(z) h(z) (z U) which shows that f(z) K, δ (; g; h) with resect to ϕ(z) Sδ, (; g; h). Corollary 2.3. Let 0 < α 1, 1 B < A 1 and ( ) α 1 + Az (29) h(z) = 1 + Bz [ If 1 ( ] ) 1 1 A α 1 B, then S δ+1, (; g; h) Sδ, (; g; h) and Kδ+1, (; g; h) K, δ (; g; h). Proof. The analytic function h(z) defined by (29) is convex univalent in U (see [8]), h(0) = 1 and h(u) is symmetric with resect to real axis. Thus h(z) P and 0 ( ) α 1 A < R{h(z)} < 1 B ( ) α 1 + A (z U; 0 < α 1; 1 < B < A 1). 1 + B Hence, by using Theorems 2.1 and 2.2 we have the corollary. Corollary 2.4. Let 0 < α 1 and (30) h(z) = 1 + 2 ( ( )) 2 1 + αz π 2 log 1 αz ( ) 2, If π2 8 arctan α then S δ+1, (; g; h) Sδ, (; g; h) and Kδ+1, (; g; h) (; g; h) K δ, Proof. The function h(z) defined by (30) is in the class P (cf. [4]) and satisfies h(z) = h(z). Therefore, R{h(z)} > h( 1) = 1 8 ( ) 2 1 arctan α π 2 (z U; 0 < α 1). 2 Hence, by Theorems 2.1 and 2.2 we have the desired result. Theorem 2.5. Let 0 α 1 < α 2. Then C δ,(α 2, ; g; h) C δ,(α 1, ; g; h). Proof. Let f(z) C, δ (α 2, ; g; h).then there exists a function ϕ(z) S, δ (; g; h) such that (31) (1 α 2 ) z(dδ,g f(z)) ϕ δ, + α (z(d,g δ f(z)) ) 2 (ϕ δ h(z), )
912 C. SELVARAJ AND K. A. SELVAKUMARAN Suose that (32) q(z) = z(dδ,g f(z)) ϕ δ,. Then q(z) is analytic in U, with q(0) = 1. Differentiating both sides of (32) we get (33) q(z) + ϕδ, (ϕ δ, (; g; q (z) = (z(dδ,g f(z)) ). z)) ) Now, using (31), (32) and (33) we deduce that (ϕ δ, (34) q(z) + w(z)zq (z) h(z), where ( z(ϕ δ, ) ) 1 w(z) = α 2 ϕ δ., In view of Lemma 1.6 and α 2 > 0, we observe that w(z) is analytic in U and R{w(z)} > 0. Consequently, in view of (34), we deduce from Lemma 1.5 that (35) q(z) h(z). Since 0 α 1 α 2 < 1 and since h(z) is convex univalent in U, we deduce from (31) and (35) that (1 α 1 ) z(dδ,g f(z)) ϕ δ, + α (z(d,g δ f(z)) ) 1 (ϕ δ, ) ((1 α 2 ) z(dδ,g f(z)) ϕ δ, + α (z(d,g δ f(z)) ) 2 (ϕ δ, ) = α 1 α 2 h(z). ) ( + 1 α ) 1 q(z) α 2 Thus f(z) C δ, (α 1, ; g; h) which comletes the roof of Theorem 2.5. Theorem 3.1. Let h(z) P and R{h(z)} > max 3. Integral oerator { 0, R(c) } (z U), where c is a comlex number such that R(c) >. If f(z) S, δ (; g; h), then the function (36) F (z) = c + z c z 0 t c 1 f(t)dt is also in the class S, δ (; g; h), rovided that F, δ 0 (0 < z < 1) where F, δ is defined as in (6).
ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS 913 Proof. Let f(z) S, δ (; g; h). Then from (36) and R(c) >, we note that F (z) A and (37) (c + )D δ,gf(z) = cd δ,gf (z) + z(d δ,gf (z)). Also, from the above, we have (38) (c + )f δ, = 1 1 ε j (cd δ,gf ) (ε j z) + εj z(dδ,gf (ε j z)) = cf δ, + z(f δ,). Let w(z) = z(f, δ ) F, δ. Then w(z) is analytic in U, with w(0) = 1, and from (38) we observe that (39) w(z) + c = (c + ) f, δ F, δ. Differentiating both sides of (39) with resect to z and using Lemma 1.6, we obtain (40) w(z) + zw (z) w(z) + c = z(f, δ ) f δ, h(z). In view of (40), Lemma 1.5 leads to w(z) h(z). If we let q(z) = z(dδ,g F (z)) F, δ, then q(z) is analytic in U, with q(0) = 1, and it follows from (37) that (41) F,(; δ g; z)q(z) = c + Dδ,gf(z) c Dδ,gF (z). Differentiating both sides of (41), we get zq (z) + z(f, δ ) q(z) = (c + ) z(dδ,g f(z)) F, δ cz(dδ,g F (z)) or equivalently, F δ, F δ, (42) zq (z) + ( w(z) + c ) q(z) = (c + ) z(dδ,g f(z)) F, δ.
914 C. SELVARAJ AND K. A. SELVAKUMARAN Now, from (39) and (42) we deduce that (43) q(z) + zq (z) w(z) + c = c + w(z) + c = z(dδ,g f(z)) f, δ h(z), z(d δ,g f(z)) F, δ because f(z) S δ,(; g; h). Combining, R{h(z)} > max{0, R(c) } and w(z) h(z) we have R{w(z) + c} > 0 Therefore, from (43) and Lemma 1.5 we find that q(z) h(z), which shows that F (z) S, δ (; g; h). By alying similar method as in Theorem 3.1, we have: Theorem 3.2. Let h(z) P and { R{h(z)} > max 0, R(c) } (z U; R(c) > ). If f(z) K, δ (; g; h) with resect to ϕ(z) Sδ, (; g; h), then the function F (z) = c + z c z belongs to the class K, δ (; g; h) with resect to G(z) = c + z c 0 z rovided that G δ, 0 (0 < z < 1). 0 t c 1 f(t)dt t c 1 g(t)dt, Acnowledgement. The authors than the referees for their valuable comments and helful suggestions. References [1] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J. 32 (1985), no. 2, 185 195. [2], Differential subordinations and inequalities in the comlex lane, J. Differential Equations 67 (1987), no. 2, 199 211. [3] S. Owa, Z. Wu, and F. Y. Ren, A note on certain subclass of Saaguchi functions, Bull. Soc. Roy. Sci. Liege 57 (1988), no. 3, 143 149. [4] F. Rønning, Uniformly convex functions and a corresonding class of starlie functions, Proc. Amer. Math. Soc. 118 (1993), no. 1, 189 196. [5] K. Saaguchi, On a certain univalent maing, J. Math. Soc. Jaan 11 (1959), 72 75. [6] C. Selvaraj and C. Santhosh Moni, Classes of analytic functions of comlex order involving a family of generalised linear oerators, Pre-rint. [7] N.-E. Xu and D.-G. Yang, Some classes of analytic and multivalent functions involving a liner oerator, Mathematical and Comuter Modelling (2008), doi: 10.1016/ j.mcm.2008.04.005 - In ress.
ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS 915 [8], An alication of differential subordinations and some criteria for starlieness, Indian J. Pure Al. Math. 36 (2005), no. 10, 541 556. [9] D.-G. Yang and J.-L. Liu, On Saaguchi functions, Int. J. Math. Math. Sci. 2003, no. 30, 1923 1931. [10] Z.-G. Wang, Y.-P. Jiang, and H. M. Srivastava, Some subclasses of multivalent analytic functions involving the Dzio-Srivastava oerator, Integral Transforms Sec. Funct. 19 (2008), no. 1-2, 129 146. Chellian Selvaraj Deartment of Mathematics Presidency College (Autonomous) Chennai-600 005, India E-mail address: amc9439@yahoo.co.in Kuathai A. Selvaumaran Deartment of Mathematics R. M. K. Engg. College Kavaraiettai-601 206, India E-mail address: selvaa1826@gmail.com