AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS

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1 AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS SUKHWINDER SINGH SUSHMA GUPTA AND SUKHJIT SINGH Department of Applied Sciences Department of Mathematics B.B.S.B. Engineering College S.L.I.E.T. Longowal Fatehgarh Sahib Punjab, India Punjab, India sushmagupta1@yahoo.com ssbilling@gmail.com sukhjit_d@yahoo.com Received: 03 May, 2009 Accepted: 05 November, 2009 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C80, 30C45. Key words: Analytic Function, Univalent function, Starlike function, Differential subordination. Page 1 of 15

2 Abstract: Acknowledgment: We show that for α 0, 2], if f A with f z) 0, z E, satisfies the condition 1 α)f z) + α 1 + zf ) z) f F z), z) then f is univalent in E, where F is the conformal mapping of the unit disk E with F 0) = 1 and { F E) = C \ w C : R w = α, I w } α2 α). Our result extends the region of variability of the differential operator 1 α)f z) + α 1 + zf ) z) f, z) implying univalence of f A in E, for 0 < α 2. The authors are thankful to the referee for valuable comments. Page 2 of 15

3 1 Introduction and Preliminaries 4 2 Main Result 7 3 Applications to Univalent Functions 9 Page 3 of 15

4 1. Introduction and Preliminaries Let H be the class of functions analytic in E = {z : z < 1} and for a C set of complex numbers) and n N set of natural numbers), let H[a, n] be the subclass of H consisting of functions of the form fz) = a+a n z n +a n+1 z n+1 +. Let A be the class of functions f, analytic in E and normalized by the conditions f0) = f 0) 1 = 0. Let f be analytic in E, g analytic and univalent in E and f0) = g0). Then, by the symbol fz) gz) f subordinate to g) in E, we shall mean fe) ge). Let ψ : C C C be an analytic function, p be an analytic function in E, with pz), zp z)) C C for all z E and h be univalent in E, then the function p is said to satisfy first order differential subordination if 1.1) ψpz), zp z)) hz), ψp0), 0) = h0). A univalent function q is called a dominant of the differential subordination 1.1) if p0) = q0) and p q for all p satisfying 1.1). A dominant q that satisfies q q for all dominants q of 1.1), is said to be the best dominant of 1.1). The best dominant is unique up to a rotation of E. Denote by S α) and Kα), respectively, the classes of starlike functions of order α and convex functions of order α, which are analytically defined as follows: and S α) = Kα) = { f A : R zf z) fz) ) } > α, z E, 0 α < 1, { f A : R 1 + zf ) } z) > α, z E, 0 α < 1. f z) We write S = S 0), the class of univalent starlike convex functions w.r.t. the origin) and K0) = K, the class of univalent convex functions. Page 4 of 15

5 A function f A is said to be close-to-convex if there is a real number α, π/2 < α < π/2, and a convex function g not necessarily normalized) such that R e iα f ) z) > 0, z E. g z) It is well-known that every close-to-convex function is univalent. In 1934/35, Noshiro [4] and Warchawski [8] obtained a simple but interesting criterion for univalence of analytic functions. They proved that if an analytic function f satisfies the condition R f z) > 0 for all z in E, then f is close-to-convex and hence univalent in E. Let φ be analytic in a domain containing fe), φ0) = 0 and R φ 0) > 0, then, the function f A is said to be φ-like in E if ) zf z) R > 0, z E. φfz)) This concept was introduced by Brickman [2]. He proved that an analytic function f A is univalent if and only if f is φ-like for some φ. Later, Ruscheweyh [5] investigated the following general class of φ-like functions: Let φ be analytic in a domain containing fe), φ0) = 0, φ 0) = 1 and φw) 0 for w fe) \ {0}. Then the function f A is called φ-like with respect to a univalent function q, q0) = 1, if zf z) φfz)) qz), z E. Let H α β) denote the class of functions f A which satisfy the condition [ R 1 α)f z) + α 1 + zf )] z) > β, z E, f z) where α and β are pre-assigned real numbers. Al-Amiri and Reade [1], in 1975, have shown that for α 0 and for α = 1, the functions in H α 0) are univalent in Page 5 of 15

6 E. In 2005, Singh, Singh and Gupta [7] proved that for 0 < α < 1, the functions in H α α) are also univalent. In 2007, Singh, Gupta and Singh [6] proved that the functions in H α β) satisfy the differential inequality R f z) > 0, z E. Hence they are univalent for all real numbers α and β satisfying α β < 1 and the result is sharp in the sense that the constant β cannot be replaced by any real number less than α. The main objective of this paper is to extend the region of variability of the operator 1 α)f z) + α 1 + zf z) f z) implying univalence of f A in E, for 0 < α 2. We prove a subordination theorem and as applications of the main result, we find the sufficient conditions for f A to be univalent, starlike and φ-like. To prove our main results, we need the following lemma due to Miller and Mocanu. Lemma 1.1 [3, p.132, Theorem 3.4 h]). Let q be univalent in E and let θ and φ be analytic in a domain D containing qe), with φw) 0, when w qe). Set Qz) = zq z)φ[qz)], hz) = θ[qz)] + Qz) and suppose that either i) h is convex, or ii) Q is starlike. In addition, assume that iii) R zh z) Qz) > 0, z E. If p is analytic in E, with p0) = q0), pe) D and ), θ[pz)] + zp z)φ[pz)] θ[qz)] + zq z)φ[qz)], then p q and q is the best dominant. Page 6 of 15

7 2. Main Result Theorem 2.1. Let α 0 be a complex number. Let q, qz) 0, be a univalent function in E such that [ ] { )} 2.1) R 1 + zq z) q z) zq z) α 1 > max 0, R qz) α qz). If p, pz) 0, z E, satisfies the differential subordination 2.2) 1 α)pz) 1) + α zp z) pz) 1 α)qz) 1) + αzq z) qz), then p q and q is the best dominant. Proof. Let us define the functions θ and φ as follows: and θw) = 1 α)w 1), φw) = α w. Obviously, the functions θ and φ are analytic in domain D = C \ {0} and φw) 0 in D. Now, define the functions Q and h as follows: and Qz) = zq z)φqz)) = α zq z) qz), hz) = θqz)) + Qz) = 1 α)qz) 1) + α zq z) qz). Page 7 of 15

8 Then in view of condition 2.1), we have 1) Q is starlike in E and 2) R z h z) > 0, z E. Qz) Thus conditions ii) and iii) of Lemma 1.1, are satisfied. In view of 2.2), we have θ[pz)] + zp z)φ[pz)] θ[qz)] + zq z)φ[qz)]. Therefore, the proof, now, follows from Lemma 1.1. Page 8 of 15

9 3. Applications to Univalent Functions On writing pz) = f z) in Theorem 2.1, we obtain the following result. Theorem 3.1. Let α 0 be a complex number. Let q, qz) 0, be a univalent function in E and satisfy the condition 2.1) of Theorem 2.1. If f A, f z) 0, z E, satisfies the differential subordination 1 α)f z) 1) + α zf z) f z) 1 α)qz) 1) + αzq z) qz), then f z) qz) and q is the best dominant. On writing pz) = zf z) fz) in Theorem 2.1, we obtain the following result. Theorem 3.2. Let α 0 be a complex number. Let q, qz) 0, be a univalent function in E and satisfy the condition 2.1) of Theorem 2.1. If f A, zf z) fz) 0, z E, satisfies the differential subordination 1 2α) zf z) fz) + α 1 + zf ) z) 1 α)qz) + α zq z) f z) qz), then zf z) fz) qz) and q is the best dominant. By taking pz) = zf z) in Theorem 2.1, we obtain the following result. φfz)) Theorem 3.3. Let α 0 be a complex number. Let q, qz) 0, be a univalent zf function in E and satisfy the condition 2.1) of Theorem 2.1. If f A, z) φfz)) 0, z E, satisfies the differential subordination 1 α) zf z) φfz)) + α 1 + zf ) z) f z) z[φfz))] 1 α)qz) + α zq z) φfz)) qz), Page 9 of 15

10 where φ is analytic in a domain containing fe), φ0) = 0, φ 0) = 1 and φw) 0 for w fe) \ {0}, then zf z) qz) and q is the best dominant. φfz)) Remark 1. When we select the dominant qz) = 1+z, z E, then 1 z and Therefore, we have Qz) = αzq z) qz) R zq z) Qz) and hence Q is starlike. We also have = 2αz 1 z 2, zq z) Qz) = 1 + z2 1 z 2. > 0, z E, 1 + zq z) q z) zq z) qz) + 1 α 1 + z2 qz) = α 1 z + 1 α 1 + z 2 α 1 z. Thus, for any real number 0 < α 2, we obtain [ R 1 + zq z) q z) zq z) qz) + 1 α ] α qz) > 0, z E. Therefore, qz) = 1+z, z E, satisfies the conditions of Theorem 3.1, Theorem 1 z 3.2 and Theorem 3.3. Moreover, 1 α)qz) 1) + α zq z) qz) z = 21 α) 1 z + 2α z 1 z = F z). 2 Page 10 of 15

11 For 0 < α 2, we see that F is the conformal mapping of the unit disk E with F 0) = 0 and { F E) = C \ w C : R w = α 1, I w } α2 α). In view of the above remark, on writing qz) = 1+z in Theorem 3.1, we have the 1 z following result. Corollary 3.4. If f A, f z) 0, z E, satisfies the differential subordination 1 α)f z) + α 1 + zf ) z) z α) f z) 1 z + 2α z 1 z, 2 where 0 < α 2 is a real number, then R f z) > 0, z E. Therefore, f is close-to-convex and hence f is univalent in E. In view of Remark 1 and Corollary 3.4, we obtain the following result. Corollary 3.5. Let 0 < α 2 be a real number. Suppose that f A, f z) 0, z E, satisfies the condition 1 α)f z) + α 1 + zf ) z) F z). f z) Then f is close-to-convex and hence univalent in E, where F is the conformal mapping of the unit disk E with F 0) = 1 and { F E) = C \ w C : R w = α, I w } α2 α). Page 11 of 15 From Corollary 3.4, we obtain the following result of Singh, Gupta and Singh [7].

12 Corollary 3.6. Let 0 < α < 1 be a real number. If f A, f z) 0, z E, satisfies the differential inequality [ R 1 α)f z) + α 1 + zf )] z) > α, f z) then R f z) > 0, z E. Therefore, f is close-to-convex and hence f is univalent in E. From Corollary 3.4, we obtain the following result. Corollary 3.7. Let 1 < α 2, be a real number. If f A, f z) 0, z E, satisfies the differential inequality [ R 1 α)f z) + α 1 + zf )] z) < α, f z) then R f z) > 0, z E. Therefore, f is close-to-convex and hence f is univalent in E. When we select qz) = 1+z 1 z Corollary 3.8. If f A, zf z) fz) 1 2α) zf z) fz) + α in Theorem 3.2, we obtain the following result. 0, z E, satisfies the differential subordination 1 + zf z) f z) where 0 < α 2 is a real number, then f S. In view of Corollary 3.8, we have the following result. ) 1 α) 1 + z 1 z + 2α z 1 z 2 = F 1z), Page 12 of 15

13 Corollary 3.9. Let 0 < α 2 be a real number. Suppose that f A, zf z) fz) 0, z E, satisfies the condition 1 2α) zf z) fz) + α 1 + zf ) z) F f 1 z). z) Then f S, where F 1 is the conformal mapping of the unit disk E with F 1 0) = 1 α and { F 1 E) = C \ w C : R w = 0, I w } α2 α). In view of Corollary 3.8, we have the following result. Corollary Let 0 < α < 1 be a real number. If f A, zf z) 0, z E, fz) satisfies the differential inequality then f S. R [ 1 2α) zf z) fz) + α 1 + zf )] z) > 0, f z) In view of Corollary 3.8, we also have the following result. Corollary Let 1 < α 2, be a real number. If f A, zf z) 0, z E, fz) satisfies the differential inequality then f S. R When we select qz) = 1+z 1 z [ 1 2α) zf z) fz) + α 1 + zf )] z) < 0, f z) in Theorem 3.3, we obtain the following result. Page 13 of 15

14 Corollary Let 0 < α 2 be a real number. Let f A, satisfy the differential subordination zf z) φfz)) 0, z E, 1 α) zf z) φfz)) +α 1 + zf ) z) f z) z[φfz))] 1 α) 1 + z φfz)) 1 z +2α z 1 z = F 1z). 2 Then zf z) 1+z, where φ is analytic in a domain containing fe), φ0) = φfz)) 1 z 0, φ 0) = 1 and φw) 0 for w fe) \ {0}. In view of Corollary 3.12, we obtain the following result. Corollary Let 0 < α 2 be a real number. Let f A, satisfy the condition 1 α) zf z) φfz)) + α zf z) φfz)) 1 + zf ) z) f z) z[φfz))] F 1 z). φfz)) 0, z E, Then f is φ-like in E, where φ is analytic in a domain containing fe), φ0) = 0, φ 0) = 1 and φw) 0 for w fe) \ {0} and F 1 is the conformal mapping of the unit disk E with F 1 0) = 1 α and { F 1 E) = C \ w C : R w = 0, I w } α2 α). Page 14 of 15

15 References [1] H.S. AL-AMIRI AND M.O. READE, On a linear combination of some expressions in the theory of univalent functions, Monatshefto für Mathematik, ), [2] L. BRICKMAN, φ-like analytic functions. I, Bull. Amer. Math. Soc., ), [3] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 225), Marcel Dekker, New York and Basel, [4] K. NOSHIRO, On the theory of schlicht functions, J. Fac. Sci., Hokkaido Univ., ), [5] St. RUSCHEWEYH, A subordination theorem for φ-like functions, J. London Math. Soc., 213) 1976), [6] S. SINGH, S. GUPTA AND S. SINGH, On a problem of univalence of functions satisfying a differential inequality, Mathematical Inequalities and Applications, 101) 2007), [7] V. SINGH, S. SINGH AND S. GUPTA, A problem in the theory of univalent functions, Integral Transforms and Special Functions, 162) 2005), [8] S.E. WARCHAWSKI, On the higher derivatives at the boundary in conformal mappings, Trans. Amer. Math. Soc., ), Page 15 of 15