n=2 In the present paper, we introduce and investigate the following two more generalized

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1 MATEMATIQKI VESNIK 59 (007), UDK originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author introduce two new subclasses S sc (k) (α, β, γ) of close-to-convex functions and C sc (k) (α, β, γ) of quasi-convex functions with respect to -symmetric conjugate points. The coefficient inequalities and integral representations for functions belonging to these classes are provided, the inclusion relationships and convolution conditions for these classes are also provided. 1. Introduction Let A denote the class of functions of the form f() = + a n n, (1.1) which are analytic in the open unit disk U = { C : < 1. Let S, S, K, C and C denote the familiar subclasses of A consisting of functions which are univalent, starlike, convex, close-to-convex and quasi-convex in U, respectively (see, for details,,4,6,7,8. Al-Amiri, Coman and Mocanu 1 once introduced and investigated a class S sc (k) of functions starlike with respect to -symmetric conjugate points, which satisfy the following inequality { f () > 0 ( U), where k is a fixed positive integer and is defined by the following equality = 1 k 1 ε ν f(ε ν ) + ε ν f(ε ν ) (ε = exp(πi/k); U). (1.) ν=0 In the present paper, we introduce and investigate the following two more generalied subclasses S sc (k) (α, β, γ) and C sc (k) (α, β, γ) of A with respect to -symmetric conjugate points, and obtain some interesting results. AMS Subject Classification: 30C45. Keywords and phrases: Close-to-convex functions, quasi-convex functions, differential subordination, Hadamard product, -symmetric conjugate points.. 65

2 66 Zhi-Gang Wang Definition 1. Let S sc (k) (α, β, γ) denote the class of functions f() in A satisfying the following inequality f () 1 β f () < 1 α, (1.3) + (1 γ) where 0 α < 1, 0 β 1, 0 γ < 1 and is defined by equality (1.). And a function f() A is in the class C sc (k) (α, β, γ) if and only if f () S sc (k) (α, β, γ). Note that S sc (k) (0, 1, 0) = S sc (k), so the class S sc (k) (α, β, γ) is a generaliation of the class S sc (k). In our proposed investigation of the classes S sc (k) (α, β, γ) and C sc (k) (α, β, γ), we shall also make use of the following lemmas. Lemma 1. 3 Let H() = 1 + n=1 h n n be analytic in U, 0 α < 1, 0 β 1 and 0 γ < 1, then the inequality H() 1 βh() + (1 γ) < 1 α ( U) can be written as H() 1 + (1 α)(1 γ) 1 (1 α)β where stands for the usual subordination. ( U), Lemma. Let 0 α < 1, 0 β 1 and 0 γ < 1, then we have S (k) sc (α, β, γ) C S. Proof. Suppose that f() S sc (k) (α, β, γ), by Lemma 1, we know that the condition (1.3) can be written as Thus we have since f () 1 + (1 α)(1 γ) 1 (1 α)β ( U). (1.4) { f () > 0 ( U) (1.5) { 1 + (1 α)(1 γ) > 0 ( U). 1 (1 α)β Now it suffices to show that S S. Substituting by ε µ (µ = 0, 1,,..., k 1) in (1.5), then (1.5) is also true, that is, { ε µ f (ε µ ) f (ε µ > 0 ( U). (1.6) )

3 Some subclasses of close-to-convex and quasi-convex functions 67 From inequality (1.6), we have { ε µ f (ε µ ) f (ε µ ) > 0 ( U). (1.7) Note that f (ε µ ) = ε µ and f (ε µ ) = ε µ, then inequalities (1.6) and (1.7) can be written as { f (ε µ ) > 0 ( U), (1.8) and { f (ε µ ) > 0 ( U). (1.9) Summing inequalities (1.8) and (1.9), we can get ( ) f (ε µ ) + f (ε µ ) > 0 ( U). (1.10) Letting µ = 0, 1,,..., k 1 in (1.10), respectively, and summing them we can get k 1 ( ) 1 f (ε µ ) + f (ε µ ) µ=0 > 0 ( U), or equivalently, { f () > 0 ( U), that is S S. This means that S (k) sc (α, β, γ) C S, hence the proof of Lemma is complete. Lemma 3. Let 0 α < 1, 0 β 1 and 0 γ < 1, then we have C (k) sc (α, β, γ) C C. Lemma 4. 5 Let 1 B B 1 < A 1 A 1, then we have 1 + A B A 1 + B. In the present paper, we shall provide the coefficient inequalities and integral representations for functions belonging to the classes S sc (k) (α, β, γ) and C sc (k) (α, β, γ), we shall also provide the inclusion relationships and convolution conditions for these classes.

4 68 Zhi-Gang Wang. Inclusion relationships We first give some inclusion relationships for the classes S sc (k) (α, β, γ) and C sc (k) (α, β, γ). Theorem 1. Let 0 β β 1 1, 0 α 1 α < 1 and 0 γ 1 γ < 1, then we have S sc (k) (α, β, γ ) S sc (k) (α 1, β 1, γ 1 ). Proof. Suppose that f() S (k) sc (α, β, γ ), by (1.4), we have f () 1 + (1 α )(1 γ ). 1 (1 α )β Since 0 α 1 α < 1, 0 β β 1 1 and 0 γ 1 γ < 1, then we have 1 (1 α 1 )β 1 (1 α )β < (1 α )(1 γ ) (1 α 1 )(1 γ 1 ) 1. Thus, by Lemma 4, we have f () 1 + (1 α )(1 γ ) 1 (1 α )β 1 + (1 α 1)(1 γ 1 ), 1 (1 α 1 )β 1 that is f() S (k) sc (α 1, β 1, γ 1 ). This means that S (k) sc (α, β, γ ) S (k) sc (α 1, β 1, γ 1 ). Corollary 1. Let 0 β β 1 1, 0 α 1 α < 1 and 0 γ 1 γ < 1, then we have C sc (k) (α, β, γ ) C sc (k) (α 1, β 1, γ 1 ). 3. Coefficient inequalities In this section, we give some coefficient inequalities for functions belonging to the classes S sc (k) (α, β, γ) and C sc (k) (α, β, γ). Theorem. Let f() = + a n n be analytic in U, if for 0 α < 1, 0 β 1 and 0 γ < 1, we have n1+(1 α)β a n + (1 α)(1 γ)+1 (a lk+1 ) (1 α)(1+β γ), (3.1) then f() S (k) sc (α, β, γ). l=1 Proof. Suppose that f() = + a n n, and is defined by equality (1.). We now let M be denoted by M := f () (1 α) βf () + (1 γ) = na n n (a n )c n n ( ) ( ) (1 α) β + na n n + (1 γ) + (a n )c n n,

5 Some subclasses of close-to-convex and quasi-convex functions 69 where c n = 1 { k 1 1, n = lk + 1, ε (n 1)ν = (ε = exp(πi/k); l N = {1,,... ). k ν=0 0, n lk + 1. (3.) Thus, for = r < 1, we have M (n a n + (a n ) c n )r n (1 α) (1 + β γ)r nβ a n + (1 γ) (a n ) c n r n ( < {n1 + (1 α)β a n + (1 α)(1 γ) + 1 (a n ) c n ) (1 α)(1 + β γ) r < {n1 + (1 α)β a n + (1 α)(1 γ) + 1 (a n ) c n (1 α)(1 + β γ) = n1 + (1 α)β a n + (1 α)(1 γ) + 1 (a lk+1 ) (1 α)(1 + β γ). l=1 From inequality (3.1), we know that M < 0, thus we can get inequality (1.3), that is f() S (k) sc (α, β, γ). This completes the proof of Theorem. Corollary. Let f() = + a n n be analytic in U, if for 0 α < 1, 0 β 1 and 0 γ < 1, we have n 1+(1 α)β a n + (1 α)(1 γ)+1(lk+1) (a lk+1 ) (1 α)(1+β γ), then f() C (k) sc (α, β, γ). l=1 4. Integral representations In this section, we provide the integral representations for functions belonging to the classes S sc (k) (α, β, γ) and C sc (k) (α, β, γ). Theorem 3. Let f() S sc (k) (α, β, γ), then we have { 1 k 1 (1 α)(1 + β γ) = exp ζ µ=0 0 ω(ε µ ζ) 1 (1 α)βω(ε µ ζ) + ω(ε µ ζ) 1 (1 α)βω(ε µ ζ) dζ, (4.1) where is defined by equality (1.), ω() is analytic in U and ω(0) = 0, ω() < 1.

6 70 Zhi-Gang Wang Proof. Suppose that f() S (k) sc (α, β, γ), by (1.4), we have f () 1 + (1 α)(1 γ)ω() =, (4.) 1 (1 α)βω() where ω() is analytic in U and ω(0) = 0, ω() < 1. Substituting by ε µ (µ = 0, 1,,..., k 1) in (4.), we have ε µ f (ε µ ) f (ε µ ) From equality (4.3), we have ε µ f (ε µ ) f (ε µ ) = 1 + (1 α)(1 γ)ω(εµ ) 1 (1 α)βω(ε µ. (4.3) ) = 1 + (1 α)(1 γ)ω(εµ ). (4.4) 1 (1 α)βω(ε µ ) Summing equalities (4.3) and (4.4), and making use of the same method as in Lemma, we have f () = 1 k (1 α)(1 γ)ω(ε µ ) µ=0 1 (1 α)βω(ε µ (1 α)(1 γ)ω(εµ ), ) 1 (1 α)βω(ε µ ) (4.5) from equality (4.5), we can get f () 1 = 1 k 1 1 µ=0 { 1 + (1 α)(1 γ)ω(ε µ ) 1 (1 α)βω(ε µ + ) (1 α)(1 γ)ω(εµ ) 1 (1 α)βω(ε µ ). (4.6) Integrating equality (4.6), we have { f () log = 1 k 1 (1 α)(1 + β γ) µ=0 0 ζ ω(ε µ ζ) 1 (1 α)βω(ε µ ζ) + ω(ε µ ζ) dζ. (4.7) 1 (1 α)βω(ε µ ζ) From equality (4.7), we can get equality (4.1) easily. This completes the proof of Theorem 3. Theorem 4. Let f() S sc (k) (α, β, γ), then we have { 1 k 1 ξ (1 α)(1 + β γ) ω(ε µ ζ) f() = exp 0 µ=0 0 ζ 1 (1 α)βω(ε µ ζ) ω(ε + µ ζ) 1 + (1 α)(1 γ)ω(ξ) dζ dξ, (4.8) 1 (1 α)βω(ε µ ζ) 1 (1 α)βω(ξ) where ω() is analytic in U and ω(0) = 0, ω() < 1.

7 Some subclasses of close-to-convex and quasi-convex functions 71 Proof. Suppose that f() S (k) sc (α, β, γ), from equalities (4.1) and (4.), we can get f () = f () 1 + (1 α)(1 γ)ω() 1 (1 α)βω() { 1 k 1 (1 α)(1 + β γ) ω(ε µ ζ) = exp µ=0 0 ζ 1 (1 α)βω(ε µ ζ) ω(ε + µ ζ) 1 + (1 α)(1 γ)ω() dζ. 1 (1 α)βω(ε µ ζ) 1 (1 α)βω() Integrating the above equality, we can get equality (4.8) easily. Hence the proof of Theorem 4 is complete. Corollary 3. Let f() C sc (k) (α, β, γ), then we have { 1 k 1 ξ (1 α)(1 + β γ) = exp ζ 0 µ=0 0 ω(ε µ ζ) 1 (1 α)βω(ε µ ζ) + ω(ε µ ζ) 1 (1 α)βω(ε µ ζ) where is defined by equality (1.), ω() is analytic in U and ω(0) = 0, ω() < 1. Corollary 4. Let f() C sc (k) (α, β, γ), then we have { 1 t 1 k 1 ξ (1 α)(1 + β γ) ω(ε µ ζ) f() = exp 0 t 0 µ=0 0 ζ 1 (1 α)βω(ε µ ζ) ω(ε + µ ζ) 1 + (1 α)(1 γ)ω(ξ) dζ dξ dt, 1 (1 α)βω(ε µ ζ) 1 (1 α)βω(ξ) where ω() is analytic in U and ω(0) = 0, ω() < Convolution conditions Finally, we provide the convolution conditions for the classes S sc (k) (α, β, γ) and sc (α, β, γ). Let f, g A, where f() is given by (1.1) and g() is defined by C (k) g() = + b n n, then the Hadamard product (or convolution) f g is defined (as usual) by (f g)() = + a n b n n = (g f)(). dζ dξ,

8 7 Zhi-Gang Wang Theorem 5. A function f() S sc (k) (α, β, γ) if and only if { { 1 f 1 (1 α)βe iθ 1 + (1 α)(1 γ)eiθ (1 ) h () 1 + (1 α)(1 γ)eiθ (f h)() 0 (5.1) for all U and 0 θ < π, where h() is given by (5.6). Proof. Suppose that f() S (k) sc (α, β, γ), we know that the condition (1.3) can be written as (1.4), since (1.4) is equivalent to f () 1 + (1 α)(1 γ)eiθ 1 (1 α)βe iθ (5.) for all U and 0 θ < π. It is easy to know that the condition (5.) can be written as 1 { 1 (1 α)βe iθ f () 1 + (1 α)(1 γ)e iθ 0. (5.3) On the other hand, it is well known that f () = f() (1 ). (5.4) And from the definition of, we know that = 1 (f h)() + (f h)(), (5.5) where h() = 1 k k 1 υ=0 1 ε υ. (5.6) Substituting (5.4) and (5.5) into (5.3), we can get (5.1) easily. This completes the proof of Theorem 5. Corollary 5. A function f() C sc (k) (α, β, γ) if and only if { { { 1 f 1 (1 α)βe iθ 1 + (1 α)(1 γ)eiθ (1 ) h () 1 + (1 α)(1 γ)eiθ f (h )() 0 for all U and 0 θ < π, where h() is given by (5.6).

9 Some subclasses of close-to-convex and quasi-convex functions 73 Acknowledgements. This work was supported by the Scientific esearch Fund of Hunan Provincial Education Department and the Hunan Provincial Natural Science Foundation (No. 05JJ30013) of People s epublic of China. The author would like to thank Prof. Chun-Yi Gao and Prof. Yue-Ping Jiang for their support and encouragement. EFEENCES 1 H. Al-Amiri, D. Coman and P.T. Mocanu, Some properties of starlike functions with respect to symmetric conjugate points, Internat. J. Math. Math. Sci. 18 (1995), P. L. Duren, Univalent Functions, Springer-Verlag, New York, C.-Y. Gao, A subclass of close-to-convex functions, J. Changsha Commun. Univ. 10 (1994), C.-Y. Gao, S.-M. Yuan and H.M. Srivastava, Some functional inequalities and inclusion relationships associated with certain families of integral operators, Comput. Math. Appl. 49 (005), M.-S. Liu, On a subclass of p-valent close-to-convex functions of order β and type α, J. Math. Study 30 (1997), K.I. Noor, On quasi-convex functions and related topics, Internat. J. Math. Math. Sci. 10 (1987), S. Owa, M.Nunokawa, H. Saitoh and H.M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 15 (00), H.M. Srivastava and S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific, Singapore, 199. (received ) School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan, People s epublic of China. higwang@163.com