General Mathematics Vol. 17, No. 4 009, 9 44 Hankel determinant for p-valently starlike and convex functions of order α Toshio Hayami, Shigeyoshi Owa Abstract For p-valently starlike and convex functions fz in the open unit disk U, the upper bounds of the functional a p+ µa p+1, defined by using the second Hankel determinant H n due to J. W. Noonan and D. K. Thomas Trans. Amer. Math. Soc. 1976, 7-46, are discussed. 000 Mathematics Subject Classification: Primary 0C45 Key words and phrases: Hankel determinant, p-valently starlike function, p-valently convex function. 1 Introduction Let A p denote the class of functions fz of the form fz = z p + a n z n p N = {1,,, } n=p+1 9
0 Toshio Hayami, Shigeyoshi Owa which are analytic in the open unit disk U = {z C : z < 1}. Furthermore, let P denote the class of functions pz of the form pz = 1 + which are analytic in U and satisfy c k z k Re pz > 0 z U. Then we say that pz P is the Carathéodory function cf. [1]. If fz A p satisfies the following condition Re zf z > α z U fz for some α 0 α < p, then fz is said to be p-valently starlike of order α in U. We denote by S pα the subclass of A p consisting of functions fz which are p-valently starlike of order α in U. Similarly, we say that fz belongs to the class K p α of p-valently convex functions of order α in U if fz A p satisfies the following inequality for some α 0 α < p. Re 1 + zf z > α z U f z As usual, in the present investigation, we write S p = S p0, K p = K p 0, S α = S 1α and Kα = K 1 α.
Hankel determinant for p-valently starlike and convex functions... 1 Remark 1. and For a function fz A p, it follows that fz K p α fz S pα if and only if if and only if z 0 zf z p S pα pfζ dζ K p α. ζ Example 1. and fz = z p 1 z p α S pα fz = z p F 1 p α, p; p + 1; z K p α where F 1 a, b; c; z represents the hypergeometric function. In [7], Noonan and Thomas stated the q th Hankel determinant as a n a n+1 a n+q 1 a H q n = det n+1 a n+ a n+q..... n, q N = {1,,, }.. a n+q 1 a n+q a n+q This determinant is discussed by several authors. For example, we can know that the Fekete and Szegö functional a a = H 1 and they consider the further generalized functional a µa, where µ is some real number see, []. Moreover, we also know that the functional a a 4 a is equivalent to H. Janteng, Halim and Darus [4] have shown the following theorems.
Toshio Hayami, Shigeyoshi Owa Theorem 1. Let fz S. Then a a 4 a 1. Equality is attained for functions fz = z 1 z = z + z + z + 4z 4 + and Theorem. fz = z 1 z = z + z + z 5 + z 7 +. Let fz K. Then a a 4 a 1 8. The present paper is motivated by these results and the purpose of this investigation is to find the upper bounds of the generalized functional a p+ µa p+1, defined by the second Hankel determinant, for functions fz in the class Spα and K p α, respectively. Preliminary results In order to discuss our problems, we need some lemmas. lemma can be found in [1] or [8]. Lemma 1. If a function pz = 1 + c k z k P, then The following c k k = 1,,,.
Hankel determinant for p-valently starlike and convex functions... The result is sharp for pz = 1 + z 1 z = 1 + z k. Using the above, we derive Lemma. If a function pz = p + c k z k satisfies the following inequality for some α 0 α < p, then Re pz > α z U 1 c k p α k = 1,,,. The result is sharp for pz = p + p αz 1 z = p + p αz k. Proof. Let qz = pz α p α = 1 + c k p α zk. Noting that qz P and using Lemma 1, we see that which implies c k p α k = 1,,, c k p α k = 1,,,.
4 Toshio Hayami, Shigeyoshi Owa Lemma. The power series for pz = 1 + c k z k converges in U to a function in P if and only if the Toeplitz determinants c 1 c c n c 1 c 1 c n 1 D n = c c 1 c n....... c n c n+1 c n+ n = 1,,,, where c k = c k, are all non-negative. They are strictly positive except for pz = m ρ k p 0 e it k z, ρk > 0, t k real and t k t j for k j, where p 0 z = 1 + z 1 z ; in this case D n > 0 for n < m 1 and D n = 0 for n m. This necessary and sufficient condition is due to Carathéodory and Toeplitz, and it can be found in []. And then, Libera and Zl/otkiewicz [5] see, also [6] have given the following result by using this lemma with n =,. Lemma 4. If a function pz P, then the representations c = c 1 + 4 c 1ζ 4c = c 1 + 4 c 1c 1 ζ 4 c 1c 1 ζ + 4 c 11 ζ η for some complex numbers ζ and η ζ 1, η 1, are obtained. By virtue of Lemma 4, we have
Hankel determinant for p-valently starlike and convex functions... 5 Lemma 5. If a function pz = p+ c k z k satisfies Re pz > α z U for some α 0 α < p, then p αc = c 1 + {4p α c 1}ζ 4p α c = c 1 + {4p α c 1}c 1 ζ {4p α c 1}c 1 ζ +p α{4p α c 1}1 ζ η for some complex numbers ζ and η ζ 1, η 1. Proof. Since qz = pz α p α = 1 + c k p α zk P, replacing c and c c by p α and c in Lemma 4, respectively, we immediately have the p α relations of the lemma. We also need the next remark. Remark. If fz Spα, then there exists a function pz = p + c k z k such that Re pz > α z U and zf z = fzpz which implies that p + na n z n p = p + n=p+1 n=p+1 l=p n a l c n l z n p where a p = 1 and c 0 = p. Therefore, we have the follwing relation n 1 n pa n = a l c n l n p + 1. l=p
6 Toshio Hayami, Shigeyoshi Owa Main results In this section, we begin with the upper bound of a p+ µa p+1 for p-valently starlike functions of order α below. Theorem. If a function fz S pα 0 α < p, then p α {p α + 1 4p αµ} µ 1 a p+ µa p+1 p α p α {4p αµ p α + 1} 1 µ p + 1 α p α µ p + 1 α p α with equality for fz = z p 1 z p α z p 1 z p α µ 1 or µ p + 1 α p α 1 µ p + 1 α. p α Proof. If fz Spα, then we have the equation which means that a p+1 = c 1 and a p+ = c + c 1. Thus, by the inequality 1 and the representation, we can suppose that c 1 = c 0 c p α without
Hankel determinant for p-valently starlike and convex functions... 7 loss of generality and we derive a p+ µa p+1 = c + c µc = 1 1 µc + c + {4p α c }ζ p α = 1 4p α {p α 4p αµ + 1}c + {4p α c }ζ Aζ. Applying the triangle inequality, we deduce Aζ 1 [ p α + 1 4p αµ c + {4p α c } ] 4p α 1 4p α [p α1 µc + 4p α ] µ p α + 1 4p α = 1 4p α [{p αµ p + 1 α}c + 4p α p α + 1 ] µ 4p α p α {p α + 1 4p αµ} µ 1, c = p α p α p α 1 p α + 1 µ, c = 0 4p α p α + 1 4p α µ p + 1 α p α, c = 0 p α {4p αµ p α + 1} µ p + 1 α, c = p α. p α
8 Toshio Hayami, Shigeyoshi Owa Equality is attained for functions fz S pα defined by zf z fz = pz = p + p αz 1 z for the case c 1 = c = p α, ζ = 1 and c = p α, or zf z fz = pz = p + p αz 1 z for the case c 1 = c = 0, ζ = 1 and c = p α. Taking α = 0 or p = 1 in Theorem, we obtain the following corollaries, respectively. Corollary 1. If a function fz S p, then p {p + 1 4pµ} µ 1 a p+ µa p+1 p p {4pµ p + 1} 1 µ p + 1 p µ p + 1 p with equality for fz = z p 1 z p z p 1 z p µ 1 or µ p + 1 p 1 µ p + 1. p
Hankel determinant for p-valently starlike and convex functions... 9 Corollary. If a function fz S α, then 1 α { α 41 αµ} µ 1 a µa 1 α 1 µ α 1 α with equality for fz = 1 α {41 αµ α} z 1 z 1 α z 1 z 1 α µ α 1 α µ 1 or µ α 1 α 1 µ α. 1 α Also, by Corollary 1 and Corollary, we readily know Corollary. If a function fz S, then 4µ µ 1 a µa 1 1 µ 1 with equality for fz = z 1 z z 1 z 4µ µ 1 µ 1 or µ 1 1 µ 1.
40 Toshio Hayami, Shigeyoshi Owa Next, in consideration of Remark 1, we derive the upper bounds of a p+ µa p+1 for p-valently convex functions. Theorem 4. If a function fz K p α 0 α < p, then a p+ µa p+1 pp α {p α + 1p + 1 4p αpp + µ} p + 1 p + µ p + 1 pp + pp α p + p + 1 pp + µ p + 1 p + 1 α pp + p α pp α {4p αpp + µ p α + 1p + 1 } p + 1 p + µ p + 1 p + 1 α pp + p α with equality for fz = z p F 1 p α, p; p + 1; z µ p + 1 pp + or µ p + 1 p + 1 α pp + p α p z p F 1, p α; 1 + p ; p + 1 z pp + µ p + 1 p + 1 α. pp + p α Proof. Noting that fz K p α if and only if
Hankel determinant for p-valently starlike and convex functions... 41 zf z p = z p + n=p+1 n p a nz n S pα and using Theorem, we see that p + p a p + 1 p+ ν a p p+1 p α {p α + 1 4p αν} p α p α {4p αν p α + 1}, that is, that a p + 1 p+ pp + νa p+1 pp α {p α + 1 4p αν} p + ν 1 pp α p + 1 ν p + 1 α p α pp α {4p αν p α + 1} p + ν p + 1 α. p α Now, putting p + 1 ν = µ, the proof of the theorem is completed. pp + When α = 0 or p = 1 in Theorem 4, the following three corollaries are obtained.
4 Toshio Hayami, Shigeyoshi Owa Corollary 4. If a function fz K p, then p {p + 1p + 1 4p p + µ} p + 1 p + µ p + 1 pp + a p+ µa p+1 p p + p + 1 pp + p + 1 µ p p + p {4p p + µ p + 1p + 1 } p + 1 p + µ p + 1 p p + with equality for z p F 1 p, p; p + 1; z fz = µ z p F 1 p, p; 1 + p ; z p + 1 p + 1 pp + pp + Corollary 5. If a function fz Kα, then 1 α { α 1 αµ} p + 1 or µ p p + p + 1 µ. p p + µ a µa 1 α 1 α {1 αµ α} µ α µ 1 α α 1 α with equality for 1 1 z α 1 α 1 fz = and 1 log µ 1 z z F 1 1, 1 α; ; z α or µ 1 α α µ. 1 α
Hankel determinant for p-valently starlike and convex functions... 4 Corollary 6. If a function fz K, then 1 µ µ a µa 1 µ 1 µ 4 µ 4 with equality for fz = 1 log z 1 z µ or µ 4 1 + z 1 z µ 4. References [1] P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 198. [] M. Fekete and G. Szegö, Eine Bemerkung uber ungerade schlichte Funktionen, J. London Math. Soc. 819, 85-89. [] U. Grenander and G. Szegö, Toeplitz Forms and their Applications, Univ. of California Press, Berkeley and Los Angeles, 1958. [4] A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. Journal of Math. Anal. 1007, 619-65.
44 Toshio Hayami, Shigeyoshi Owa [5] R. J. Libera and E. J. Zl/otkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85198, 5-0. [6] R. J. Libera and E. J. Zl/otkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87198, 51-57. [7] J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 1976, 7-46. [8] Ch. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975. Toshio Hayami Kinki University Department of Mathematics Higashi-Osaka, Osaka 577-850, Japan e-mail: ha ya to11@hotmail.com Shigeyoshi Owa Kinki University Department of Mathematics Higashi-Osaka, Osaka 577-850, Japan e-mail: owa@math.kindai.ac.jp