The Relationships Between p valent Functions and Univalent Functions

Transcription

1 DOI:.55/auom-25-5 An. Şt. Univ. Ovidius Constanţa Vol. 23(),25, The Relationships Between p valent Functions and Univalent Functions Murat ÇAĞLAR Abstract In this paper, we obtain some sufficient conditions for general p valent integral operators to be the p th power of a univalent functions in the open unit disk. Introduction Let A(p) be the class of functions of the form: f(z) = z p + a p+n z p+n (p N = {, 2,...}) () n= which are analytic and p valent in the unit disc U = {z : z < }. A function f(z) A(p) is called p valent starlike of order γ if f(z) satisfies { zf } (z) R > γ (2) f(z) for γ < p, p N and z U. By S (p, γ), we denote the class of all p valent starlike functions of order γ. By S p (γ) denote the subclass of S (p, γ) consisting of functions f(z) A(p) for which zf (z) f(z) p < p γ (3) Key Words: Univalent functions, p valent functions, p valent integral operators. 2 Mathematics Subject Classification: Primary 3C45; Secondary 3C55, 3C8. Received: 5 April, 24. Revised: June, 24. Accepted: 28 June,

2 UNIVALENT FUNCTIONS 66 for γ < p, p N and z U. Also a function f(z) A(p) is called p valent convex of order γ if f(z) satisfies { R + zf } (z) f > γ (4) (z) for γ < p, p N and z U. By C (p, γ), we denote the class of all p valent convex functions of order γ. It follows from (2) and (4) that f(z) C (p, γ) zf (z) p S (p, γ). (5) Also C p (γ) denote the subclass of C (p, γ) consisting of functions f(z) A(p) for which zf (z) f (p ) (z) < p γ (6) for γ < p, p N and z U. We define the following general p valent integral operators: The first family of p valent integral operators has the following form: F p α,...,α n,β (z) = {βp n u βp ( ) } /αi /β fi (u) du, (7) where the functions f i for all i =, n, n N = {, 2,...} belongs to the class A(p) and the parameters β and α i (α i ) for all i =, n are complex numbers such that the integral operators in (7) exist. Remark : (7) p valent integral operator, for p =, was studied by Seenivasagan and Breaz (see [4]) (see also the recent investigations on this subject by Baricz and Frasin [2] and Srivastava, Deniz and Orhan [5]). We note that if α i = α for all i =, n and p =, then the p valent integral operator F p α (z) reduces to the operator,...,α n,β F α,β (z) which is related closely to some known integral operators investigated earlier in Geometric Functions Theory (see, for details, [6]). The operators Fα,β (z) and F α,α(z) were studied by Breaz and Breaz (see [4]) and Pescar (see [2]), respectively. Upon setting β = and α = β = in Fα,β (z), we can obtain the operators F α,(z) and F,(z) which were studied by Breaz and Breaz (see [3]) and Alexander (see []), respectively. Furthermore, in their special cases when p = n = β =, and /α instead of α i = α for all i =, n, the p valent integral operator in (7) would obviously reduce to the operator F/α, (z) which was studied Pescar and Owa (see [3]), for α [, ] special case of the operator F/α, (z) was studied by Miller, Mocanu and Reade (see []). Recently, Bulut (see u p

3 UNIVALENT FUNCTIONS 67 [6]) introduced this operator for p = by using the generalized Al-Oboudi differential operator. The second family of p valent integral operators has the following form: G p α,...,α n,β (z) = {βp n u βp ( ) } f αi /β i (u) pu p du (8) where the functions f i A(p) for all i =, n and the parameters β and α i for all i =, n are complex numbers such that the p valent integral operators in (8) exist. Remark 2: For p =, G α,...,α n,β (z) was introduced by Breaz and Breaz (see [5]). Additionally, for β =, G p α,...,α n, (z) was studied by Frasin (see [9]). Hallenbeck and Livingston (see []) defined p subordination chains and they obtained some results for f A(p) to be the p th power of a univalent functions in U. Recently, Deniz (see [7]) investigated some results for f A(p) to be the p th power of a univalent functions in U by using p subordination chains method. Also, Deniz et al. (see [8]) submitted a paper which includes sufficient conditions for a p valent integral operator to be the p th power of a univalent function in U by using p subordination chains method. In our present investigation, we need one of sufficient conditions which we recall here as Theorem. below. This theorem is of fundemental importance to our investigation. Theorem.. ([8]) Let f A(p) and α complex number such that R(α) >. Suppose that z 2pR(α) zf (z) R(α) f (p ) (z) p (9) is true for all z U, then the integral operator H α (z) = α u p(α ) f (u)du α () 2 Main Results Firstly, we obtain sufficient conditions for (7) p valent integral operator to be the p th power of a univalent function in U.

4 UNIVALENT FUNCTIONS 68 Theorem 2.. Let f i S (p, γ i ) ( γ i < p) for all i =, n. If the parameters β and α i (α i ) for all i =, n are complex numbers and pr (β) p + n (p γ i ) α i () then the integral operator F p α,...,α n,β defined by (7) is the p th power of a univalent function in U where the principal branch is Proof. Define the function It is easy to see that h(z) = h (z) = n ( ) /αi fi (u) du. (2) u p n ( ) /αi fi (z). (3) Differentiating (3) logarithmically and multiplying by z, we obtain zh (z) h (z) = n α i In the light of the hypothesis of Theorem 2., we have z p ( ) zf i (z) f i (z) p. (4) z 2pR(β) R(β) p + zh (z) h (z) = z 2pR(β) n R(β) p + ( ) zf i (z) α i f i (z) p [ n z 2pR(β) p + zf i (z) ] R(β) α i f i (z) p [ ] n p + R(β) α i (p γ i) p. Hence by Theorem., we get the integral operator F p α,...,α n,β defined by (7) This completes the proof. Letting β = n =, following corollary. α = α, γ = γ and f = f in Theorem 2., we obtain

5 UNIVALENT FUNCTIONS 69 Corollary 2.2. Let f S (p, γ). If the parameter α ( α 2) is complex number and p α γ (5) 2 α then the integral operator F p α(z) = ( ) /α f(u) pu p u p du (6) By taking α = in Corollary 2.2, we get: Corollary 2.3. Let f S (p, γ). If then the integral operator p γ (7) F p (z) = p f(u) du (8) u In the next theorem, we derive another sufficient condition for (8) p valent integral operator to be the p th power of a univalent function in U. Theorem 2.4. Let f i C (p, γ i ) ( γ i < p) for all i =, n. If the parameters β and α i for all i =, n are complex numbers and pr (β) p + n α i (p γ i ) (9) then the integral operator G p α,...,α n,β defined by (8) is the p th power of a univalent function in U where the principal branch is Proof. We define It is easy to see that g(z) = g (z) = n n ( ) f αi i (u) pu p du. (2) ( ) f αi i (z) pz p. (2)

6 UNIVALENT FUNCTIONS 7 Differentiating (2) logarithmically and multiplying by z, we obtain zg (z) n g (z) = ( zf i α (z) ) i (p ). (22) (z) In the light of the hypothesis of Theorem 2.4, we have f i z 2pR(β) R(β) p + zg (z) g (z) = z 2pR(β) n R(β) p + ( zf i α (z) ) i f (p ) i (z) [ n z 2pR(β) p + α i zf i (z) ] R(β) f (p ) i (z) [ ] n p + α i (p γ i ) p. R(β) Consequently, by Theorem., we get the integral operator G p α,...,α n,β defined by (8) is the p th power of a univalent function in U where the principal branch is Thus, the proof is completed. For β = n =, α = α, γ = γ and f = f in Theorem 2.4, we get following corollary. Corollary 2.5. Let f C (p, γ). If the parameter α ( α 2) is complex number and p γ α (23) 2 α then the integral operator G p α(z) = ( f pu p ) α (u) pu p du (24) If we choose α = in Corollary 2.5, we have: Corollary 2.6. Let f C (p, γ). If p γ (25)

7 UNIVALENT FUNCTIONS 7 then the integral operator G p (z) = f (u)du (26) References [] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. (Ser. 2) 7 (95), [2] Á. Baricz and B. A. Frasin, Univalence of integral operators involving Bessel functions, Appl. Math. Lett. 23 (2), [3] D. Breaz and N. Breaz, Two integral operators, Stud. Univ. Babeş-Bolyai. Math. 47 (3) (22), 3 9. [4] D. Breaz and N. Breaz, The univalent conditions for an integral operator on the classes S p and T 2, J. Approx. Theory Appl. (25), [5] D. Breaz and N. Breaz, Univalence conditions for certain integral operators, Stud. Univ. Babeş-Bolyai Math. 47 (2) (22), 9 5. [6] S. Bulut, Univalence preserving integral operators defined by generalized Al-Oboudi differential operators, An. Şt. Univ. Ovidius Constanta. Vol. 7() (29), [7] E. Deniz, p-subordination chains and p-valence criteria, Journal of Inequalities and Applications. 23. (23), -7. [8] E. Deniz, H. Orhan, M. Çağlar, Sufficient conditions for p valence of an integral operator (submitted). [9] B. A. Frasin, New general integral operators of p-valent functions, Journal of Inequatilies Pure and Applied Mathematics (4) (29), 9 pp. [] D. J. Hallenbeck and A. E. Livingston, Subordination chains and p valent functions, preprint 975. [] S. S. Miller, P. T. Mocanu, and M. O. Reade, Starlike integral operators, Pacific J. Math. 79 (978), [2] V. Pescar, New criteria for univalence of certain integral operators, Demonstratio Math. 33 (2), 5 54.

8 UNIVALENT FUNCTIONS 72 [3] V. Pescar and S.Owa, Sufficient conditions for univalence of certain integral operators, Indian J. Math. 42 (2), [4] N. Seenivasagan and D. Breaz, Certain sufficient conditions for univalence, Gen. Math. 5 (4) (27), 7 5. [5] H. M. Srivastava, E. Deniz and H. Orhan, Some general univalence criteria for a family of integral operators, Appl. Math. Comput. 25 (2), [6] H. M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 992. Murat ÇAĞLAR, Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36, Kars, Turkey. mcaglar25@gmail.com