International Journal of Mathematical Analysis Vol. 9, 05, no. 0, 493-498 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.55 Second Hankel Determinant Problem for a Certain Subclass of Univalent Functions Tugba Yavuz Gebze Technical University Department of Mathematics Gebze, Kocaeli, Turkey Copyright c 05 Tugba Yavuz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let S denote the class of analytic and univalent functions in the open unit disk D = {z : z < with the normalization conditions. In the present artical an upper bound for the second Hankel determinant a a 4 a 3 is obtained for a certain subclass of univalent functions. Mathematics Subject Classification: Primary 30C45 Keywords: Univalent Functions, Second Hankel Determinant Introduction Let D be the unit disk {z : z <, A be the class of functions analytic in D, satisfying the conditions Then each function f f0) = 0 and f 0) =. ) in A has the Taylor expansion fz) = z + a n z n ) because of the conditions ). Let S denote class of analytic and univalent functions in D with the normalization conditions ). n=
494 T. Yavuz as The q th determinant for q and n 0 is stated by Noonan and Thomas a n a n+ a n+q+ a n+... H q n) =. 3).. a n+q a n+q This determinant has also been considered by several authors. For example, Noor in [] determined the rate of growth of H q n) as n for functions f given by ) with bounded boundary. Ehrenborg in [] stadied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some of its properties were discussed by Layman s article [9]. It is well known that [] that for f S and given by ) the sharp inequality a 3 a holds. This corresponds to the Hankel determinant with q = and k =. For a given class of functions in A, the sharp bound for the nonlinear functional a a 4 a 3 is known as the second Hankel determinant. This corresponds to the Hankel determinant with q = and k =. Janteng, Halim and Darus [8] have considered the functional a a 4 a 3 and found a sharp bound for the function f in the subclasses S and C, starlike and convex functions as follows. Theorem Let f S. Then. 4) This result is sharp. Theorem Let f C. Then This result is sharp. 8. 5) In particular, sharp bounds on H ) were obtained by several authors of articles [7], [3], [0], [5] and [6] for different subclasses of univalent functions. In this paper, we introduce a certain subclass of analytic functions and obtain an upper bound to the functional a a 4 a 3 for the function f belonging to this class, defined as follows. Definition Let λ [0, ) and let f be an univalent function of the form ). We say that f belongs to F λ) if and only if ) zf z) Re > 0, z D. 6) λ) f z) + λzf z)
Hankel Determinant 495 Preliminary Results The following lemmas are required to prove our main results. Let P be the family of all functions p analytic in D for which Re pz)) > 0 and pz) = + c z + c z +. 7) Lemma 3 Duren, []) If p P, then c k for each k N. Lemma 4 Grenander&Szegö [4]) The power series for pz) given by 6) converges in D to a function in P if and only if the Toeplitz determinants c c c n c c c n D n =, n =,,. 8)..... c n c n+ c n+ and c k = c k, are all nonnegative. They are strictly positive except for pz) = m ρ k p 0 e itkz ), ρ k > 0, t k real and t k t j for k j; in this case D n > 0 for k= n < m and D n = 0 for n m. We may assume that without restriction that c > 0. On using Lemma., for n = and n = 3 respectively, we get c c D = c c c c = 8 + Re { c c c 4c 0, which is equivalent to c = c + x ) 4 c for some x, x. If we consider the determinant c c c 3 D n = c c c c c c 0, c 3 c c we get the following inequality ) 4c 3 4c c + c) 3 4 c + c c c) ) 4 c c c). From 9) and 0), it is obtained that 9) 0) 4c 3 = c 3 + c 4 c ) x c 4 c ) x + c 4 c ) x ) z ) for some z, z.
496 T. Yavuz 3 Main Results Now, we are ready to prove our main result. Theorem 5 Let the function f given by ) be in the class in F λ). Then 33λ 3 + 36λ + λ + 3 3 λ) 5. Proof. Since f F λ), there exists an analytic function p P in the unit disk D with p0) = and Re pz)) > 0 such that zf z) λ) f z) + λzf z) = pz) ) for some z D. By using the series expansions of fz) and pz) as in ) and 7), equating coefficients in ) yields c a = λ a 3 = { c λ + + λ λ) c a 4 = Hence, we get from 3) a a 4 a 3 = 6λ + λ ) c 3 λ + 4λ + 3 6 λ) c + λ) c 6 λ) 4 c 3. 3) c c 3 3 λ) + λ λ) 3 c c 4) + λ { 9λ λ) 5 + λ + c 4 c 4 λ). Since the function pz) and pe iθ z), θ R) are in the class P simultaneously, we assume that without loss of generality that c > 0. For convenience of notation, we take c = c, c [0, ]. Using 0) and ) in 4), we obtain 3 = 33λ 3 + 36λ + λ + 3 48 λ) 5 c 4 + + λ 4 λ) 3 xc 4 c ) + c 4 c ) x ) z 6 λ) 4 c ) x + c ) 48 λ). Application of the triangle inequality gives { 33λ 3 + 36λ + λ + 3 48 λ) λ) 3 c 4 5) +8c 4 c ) ρ ) ρ + λ) + λ c 4 c ) + 4 c ) ρ + c ) = Gc, ρ)
Hankel Determinant 497 where x = ρ and c [0, ]. We now maximize the function Gc, ρ) on the closed square [0, ] [0, ]. Since Gc, ρ) ρ and Gc,ρ) ρ = { 4 λ) ρ 4 c ) + λ) c ) c 6) + λ c 4 c ) > 0, Gc, ρ) can not have a maximum in the interior of the closed square [0, ] [0, ]. Hence, for fixed c [0, ] Since F c) = max Gc, ρ) = Gc, ) = F c). 0 ρ { 33λ 3 + 36λ + λ + 3 48 λ) λ) 3 c 4 + λ) + λ c 4 c ) + 4 c ) + c ), after elemantary calculus, one can show that F c) = c λ) { 33λ 3 + 36λ + λ + 3 λ) 3 c 3 + 8λ λ is positive. So, F c) is an increasing function. Therefore, the upper bounds of 5) corresponds to ρ = and c =. Hence 33λ 3 + 36λ + λ + 3 3 λ) 5. The proof is completed. For λ = 0, we get the following result. Corollary 6 If f S then. This result is sharp. References [] P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 983. [] R. Ehrenborg, The Hankel determinant of exponantial polynomials, American Mathematical Monthly, 07 000), 557-560. http://dx.doi.org/0.307/58935
498 T. Yavuz [3] M. Fekete and G. Szegö, Eine Bemerkung uber ungerade schlichte Funktionen, J. London Math. Soc, 8 933), 85-89. http://dx.doi.org/0./jlms/s-8..85 [4] U. Grenander and G. Szegö, Toeplitz forms and their application, Univ. of Calofornia Press, Berkely and Los Angeles, 958). [5] T. Hayami and S. Owa, Hankel determinant for p-valently starlike and convex functions of order α, General Math., 7 009), 9-44. [6] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4 00), 573-585. [7] A. Janteng, S. A. Halim, and M. Darus, Coefficient inequality for a function whose derivative has positive real part, J. Ineq. Pure and Appl. Math, 7 ) 006), -5. [8] A. Janteng, Halim, S. A. and Darus, M. : Hankel Determinant For Starlike and Convex Functions, Int. Journal of Math. Analysis, I 3) 007), 69-65. [9] J. W. Layman, The Hankel transform and some of its properties. J. of integer sequences, 4 00), -. [0] G. Murugusundaramoorthy and N. Magesh, Coefficient Inequalities For Certain Classes of Analytic Functions Associated with Hankel Determinant, Bulletin of Math. Anal. Appl., I 3) 009), 85-89. [] J. W. Noonan and D. K. Thomas, On the second Hankel Determinant of a really mean p valent functions, Trans. Amer. Math. Soc, 3 ) 976), 337-346. http://dx.doi.org/0.090/s000-9947-976-04607-9 [] K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl, 8 8) 983), 73-739. [3] S. C. Soh and D. Mohamad, Coefficient Bounds For Certain Classes of Close-to-Convex Functions, Int. Journal of Math. Analysis, 7) 008), 343-35. Received: January 6, 05; Published: February 3, 05