On Starlike and Convex Functions with Respect to 2k-Symmetric Conjugate Points

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1 Tamsui Oxford Journal of Mathematical Sciences 24(3) (28) Aletheia University On Starlike and Convex Functions with espect to 2k-Symmetric Conjugate Points Zhi-Gang Wang and Chun-Yi Gao College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 476 Hunan, People s epublic of China eceived February 27, 27, Accepted March 27, 27. Abstract In the present paper, two new classes S (k) sc (φ) and C (k) sc (φ) of functions starlike and convex with respect to 2k-symmetric conjugate points are introduced. The integral representations for functions belonging to these classes are provided, the convolution conditions, growth, distortion and covering theorems for these classes are also provided. Keywords and Phrases: Starlike functions, Convex functions, Close-toconvex functions, Quasi-convex functions, Differential subordination, Hadamard product, 2k-symmetric conjugate points.. Introduction Let A denote the class of functions of the form f() = + a n n, (.) n=2 2 Mathematics Subject Classification. Primary 3C45. higwang@63.com cygao@63.com

2 278 Zhi-Gang Wang and Chun-Yi Gao which are analytic in the open unit disk U = { C : < }. Let S, S, K, C and C denote the familiar subclasses of A consisting of functions which are, respectively, univalent, starlike, convex, close-to-convex and quasi-convex in U (see, for details, 3, 5, 6, 9). Also let P denote the class of functions of the form p() = + p n n ( U), n= which satisfy the condition {p()} >. Let f() and F () be analytic in U. Then we say that the function f() is subordinate to F () in U, if there exists an analytic function ω() in U such that ω() and f() = F (ω()), denoted by f F or f() F (). If F () is univalent in U, then the subordination is equivalent to f() = F () and f(u) F (U) (see 7). A function f() A is in the class S (φ) if f() satisfies the condition f () f() φ() ( U), where φ() P. The class S (φ) and a corresponding convex class K(φ) were defined by Ma and Minda 4. And the results about the convex class K(φ) can be easily obtained from the corresponding results of functions in S (φ). A function f() A is in the class S sc(φ) if f() satisfies the condition 2f () f() f( ) φ() ( U), where φ() P. And a function f() A is in the class C sc (φ) if and only if f () Ssc(φ). The classes Ssc(φ) of functions starlike with respect to symmetric conjugate points and C sc (φ) of functions convex with respect to symmetric conjugate points were considered recently by avichandran 8. Furthermore, Chen, Wu and Zou 2 discussed a class of functions α-starlike with respect to symmetric conjugate points. A function f() A is in the class S s (k) (φ) if f() satisfies the condition f () f k () φ() ( U),

3 On Starlike and Convex Functions 279 where φ() P, k 2 is a fixed positive integer and f k () is defined by the following equality f k () = k ε ν f(ε ν ) (ε = exp(2πi/k); U). k ν= And a function f() A is in the class C (k) s The classes S (k) s C (k) s (φ) if and only if f () S s (k) (φ). (φ) of functions starlike with respect to k-symmetric points and (φ) of functions convex with respect to k-symmetric points were considered recently by Wang, Gao and Yuan. Al-Amiri, Coman and Mocanu once introduced and investigated a class of functions starlike with respect to 2k-symmetric conjugate points, which satisfy the following inequality { f () f 2k () } > ( U), where k 2 is a fixed positive integer and f 2k () is defined by the following equality f 2k () = 2k k ν= ε ν f(ε ν ) + ε ν f(ε ν ) (ε = exp(2πi/k); U). (.2) But until now, we can not give the definition of functions starlike with respect to k-conjugate points (k 3), this is still an unsolved problem. Motivated by the above mentioned classes, we now introduce the following two classes of functions starlike and convex with respect to 2k-symmetric conjugate points, and obtain some interesting results. Definition. A function f() A is in the class S sc (k) (φ) if f() satisfies the condition f () φ() ( U), (.3) f 2k () where φ() P and f 2k () is defined by the equality (.2). And a function f() A is in the class C sc (k) (φ) if and only if f () S sc (k) (φ). In the present paper, we shall provide the integral representations for functions belonging to the classes S sc (k) (φ) and C sc (k) (φ), we shall also provide the convolution conditions, growth, distortion and covering theorems for these classes.

4 28 Zhi-Gang Wang and Chun-Yi Gao 2. Integral epresentations We first give some inclusion relationships for the classes S sc (k) (φ) and C sc (k) (φ), which tell us that S sc (k) (φ) is a subclass of close-to-convex functions, and C sc (k) (φ) is a subclass of quasi-convex functions. Theorem. Let φ() P, then we have S sc (k) (φ) C S. Proof. Suppose that f() S sc (k) (φ), it suffices to show that f 2k () S S. From the condition (.3), we have { } f () > ( U) (2.) f 2k () since {φ()} >. Substituting by ε µ (µ =,, 2,..., k ) in (2.), then (2.) is also true, that is, { } ε µ f (ε µ ) > ( U). (2.2) f 2k (ε µ ) From inequality (2.2), we have { } ε µ f (ε µ ) f 2k (ε µ ) > ( U). (2.3) Note that f 2k (ε µ ) = ε µ f 2k () and f 2k (ε µ ) = ε µ f 2k (), then inequalities (2.2) and (2.3) can be written as { } f (ε µ ) > ( U), (2.4) f 2k () and { } f (ε µ ) > ( U). (2.5) f 2k () Summing inequalities (2.4) and (2.5), we can get f (ε µ ) + f (ε µ ) > ( U). (2.6) f 2k ()

5 On Starlike and Convex Functions 28 Let µ =,, 2,..., k in (2.6), respectively, and summing them we can get ( ) k 2k µ= f (ε µ ) + f (ε µ ) > ( U), f 2k () or equivalently, { } f 2k () > ( U), f 2k () that is f 2k () S S. This means that S sc (k) (φ) C S, and hence the proof of Theorem is complete. Similarly, for the class C sc (k) (φ), we have Corollary. Let φ() P, then we have C sc (k) (φ) C C. We now provide the integral representations for functions belonging to the classes S sc (k) (φ) and C sc (k) (φ). Theorem 2. Let f() S sc (k) (φ), then we have { k } f 2k () = exp 2k ζ µ ζ)) 2 dζ, (2.7) µ= where f 2k () is defined by equality (.2), ω() is analytic in U and ω() =, ω() <. Proof. Suppose that f() S sc (k) (φ), from the definition of S sc (k) (φ), we have f () f 2k () = φ(ω()), (2.8) where ω() is analytic in U and ω() =, ω() <. ε µ (µ =,, 2,..., k ) in (2.8), we have Substituting by ε µ f (ε µ ) f 2k (ε µ ) = φ(ω(ε µ )). (2.9) From equality (2.9), we have ε µ f (ε µ ) = φ(ω(ε µ )). (2.) f 2k (ε µ )

6 282 Zhi-Gang Wang and Chun-Yi Gao Summing equalities (2.9) and (2.), and making use of the same method as in Theorem, we have f 2k () f 2k () = 2k from equality (2.), we can get f 2k () f 2k () = 2k k µ= k µ= φ(ω(ε µ )) + φ(ω(ε µ )), (2.) φ(ω(ε µ )) + φ(ω(ε µ )) 2. (2.2) Integrating equality (2.2), we have { } f2k () log = k 2k ζ µ ζ)) 2 dζ. (2.3) µ= From equality (2.3), we can get equality (2.7) easily. Hence the proof is complete. Theorem 3. Let f() S sc (k) (φ), then we have { k } ξ f() = exp 2k ζ µ ζ)) 2 dζ µ= φ(ω(ξ))dξ, (2.4) where ω() is analytic in U and ω() =, ω() <. Proof. Suppose that f() S sc (k) (φ), from equalities (2.7) and (2.8), we can get { f () = f 2k() k } φ(ω()) = exp 2k ζ µ ζ)) 2 dζ φ(ω()). µ= (2.5) Integrating equality (2.5), we can get equality (2.4) easily. Hence the proof is complete. Similarly, for the class C sc (k) (φ), we have Corollary 2. Let f() C sc (k) (φ), then we have { k } ξ f 2k () = exp 2k ζ µ ζ)) 2 dζ dξ, µ=

7 On Starlike and Convex Functions 283 where f 2k () is defined by equality (.2), ω() is analytic in U and ω() =, ω() <. Corollary 3. Let f() C sc (k) (φ), then we have { t k } ξ f() = exp t 2k ζ µ ζ)) 2 dζ φ(ω(ξ))dξdt, µ= where ω() is analytic in U and ω() =, ω() <. 3. Convolution Conditions In this section, we give the convolution conditions for the classes S sc (k) (φ) and C sc (k) (φ). Let f, g A, where f() is given by (.) and g() is defined by g() = + b n n, then the Hadamard product (or convolution) f g is defined (as usual) by (f g)() = + n=2 a n b n n = (g f)(). n=2 Theorem 4. Let f() A and φ() P, then f() S sc (k) (φ) if and only if ( ) f ( ) φ(eiθ ) h () φ(eiθ ) (f h)() (3.) for all U and θ < 2π, where h() is given by (3.6). Proof. Suppose that f() S (k) sc (φ), since f () f 2k () φ() if and only if f () f 2k () φ(eiθ ) (3.2)

8 284 Zhi-Gang Wang and Chun-Yi Gao for all U and θ < 2π. And the condition (3.2) can be written as On the other hand, it is well known that f () f 2k ()φ(e iθ ). (3.3) f () = f() And from the definition of f 2k (), we know where f 2k () = 2 ( ) 2. (3.4) (f h)() + (f h)(), (3.5) h() = k k υ= ε υ. (3.6) Substituting (3.4) and (3.5) into (3.3), we can get (3.) easily. This completes the proof of Theorem 4. Similarly, for the class C sc (k) (φ), we have Corollary 4. Let f() A and φ() P, then f() C sc (k) (φ) if and only if { ( ) } f ( ) φ(eiθ ) h () φ(eiθ ) f (h )() for all U and θ < 2π, where h() is given by (3.6). 4. Growth, Distortion and Covering Theorems Finally, we provide the growth, distortion and covering theorems for the classes S sc (k) (φ) and C sc (k) (φ). For the purpose of this section, assume that the function φ() is an analytic function with positive real part in the unit disk U, φ(u) is convex and symmetric with respect to the real axis, φ() = and φ () >. The functions k φn () (n = 2, 3,...) defined by k φn () = k φn () = and + k φn () k φn () = φ(n )

9 On Starlike and Convex Functions 285 are important examples of functions in K(φ). The functions h φn () satisfying h φn () = k φn () are examples of functions in S (φ). Write k φ2 () simply as k φ () and h φ2 () simply as h φ (). Note that if f() S sc (k) (φ), then f 2k () S (φ). Therefore, by making use of the similar method as in Theorem 7 obtained by avichandran 8, we can get the following theorem, here we omit the details. Theorem 5. Let min =r φ() = φ( r), max =r φ() = φ(r), = r <. If f() S (k) sc (φ), then we have and h φ( r) f () h φ(r), h φ ( r) f() h φ (r), f(u) {ω : ω h( )}. These results are sharp. Similarly, note that if f() C (k) sc (φ), then f 2k () = + l= a lk+ + a lk+ lk+ = + 2 (a lk+ ) lk+ K(φ). Therefore, by making use of the similar method as in Theorem 9 obtained by Wang, Gao and Yuan, we can get the following theorem, here we also omit the details. Theorem 6. Let min =r φ() = φ( r), max =r φ() = φ(r), = r <. If f() C sc (k) (φ), then we have r s r s r φ( t)k φ( t k ) /k dt f () r φ( t)k φ( t k ) /k dtds f() and { f(u) ω : ω s These results are sharp. s l= r r s φ(t)k φ(t k ) /k dt, s φ(t)k φ(t k ) /k dtds, } φ( t)k φ( t k ) /k dtds.

10 286 Zhi-Gang Wang and Chun-Yi Gao Acknowledgements This work was supported by the Scientific esearch Fund of Hunan Provincial Education Department and the Hunan Provincial Natural Science Foundation (No. 5JJ33) of People s epublic of China. The authors would like to thank Prof. H.M. Srivastava for his support and encouragement. eferences 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. 8 (995), M.-P. Chen, Z.-. Wu and Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 2 (996), P. L. Duren, Univalent Functions, Springer-Verlag, New York, W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in: Proc. Conf. Complex Analysis, Tianjin, 992, in: Conf. Proc. Lecture Notes Anal., I, Internat. Press, Cambridge, MA, 994, pp K. I. Noor, On quasi-convex functions and related topics, Internat. J. Math. Math. Sci. (987), S. Owa, M.Nunokawa, H. Saitoh and H.M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 5 (22), C. Pommerenke, Univalent Functions, Vandenhoeck and uprecht, Göttingen, V. avichandran, Starlike and convex functions with respect to conjugate points, Acta Math. Acad. Paedagog. Nyhái. (N.S.) 2 (24), H. M. Srivastava and S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific, Singapore, 992.

11 On Starlike and Convex Functions 287 Z.-G. Wang, C.-Y. Gao and S.-M. Yuan, On certain subclasses of closeto-convex and quasi-convex functions with respect to k-symmetric points, J. Math. Anal. Appl. 322 (26), 97-6.