General Mathematics Vol. 13, No. 3 (2005), 23 30 On a subclass of n-close to convex functions associated with some hyperbola Mugur Acu Dedicated to Professor Dumitru Acu on his 60th anniversary Abstract In this paper we define a subclass of n-close to convex functions associated with some hyperbola and we obtain some properties regarding this class. 2000 Mathematics Subject Classification: 30C45 Key words and phrases: n-close to convex functions, Libera-Pascu integral operator, Briot-Bouquet differential subordination 1 Introduction Let H(U) be the set of functions which are regular in the unit disc U = {z C : z < 1}, A = {f H(U) : f(0) = f (0) 1 = 0} and S = {f A : f is univalent in U}. We recall here the definition of the well - known class of close to convex functions: CC = { f A : exists g S, Re zf (z) g(z) 23 } > 0, z U.
24 Mugur Acu Let consider the Libera-Pascu integral operator L a : A A defined as: (1) f(z) = L a F (z) = 1 + a z a z 0 F (t) t a 1 dt, a C, Re a 0. For a = 1 we obtain the Libera integral operator, for a = 0 we obtain the Alexander integral operator and in the case a = 1, 2, 3,... we obtain the Bernardi integral operator. Let D n be the S al agean differential operator (see [6]) D n : A A, n N, defined as: D 0 f(z) = f(z) D 1 f(z) = Df(z) = zf (z) D n f(z) = D(D n 1 f(z)) We observe that if f S, f(z) = z + a j z j, z U then D n f(z) = z + j n a j z j. The purpose of this note is to define a subclass of n-close to convex functions associated with some hyperbola and to obtain some estimations for the coefficients of the series expansion and some other properties regarding this class. 2 Preliminary results Definition 1. (see [7] ) A function f S is said to be in the class SH(α) if it satisfies zf (z) f(z) 2α ( 2 1 ) < Re for some α (α > 0) and for all z U. { } 2 zf (z) ( ) + 2α 2 1, f(z)
On a subclass of n-close to convex functions... 25 Definition 2. (see [2]) Let f S and α > 0. We say that the function f is in the class SH n (α), n N, if D n+1 f(z) ( ) 2α 2 1 < Re D n f(z) { } 2 D n+1 f(z) ( ) +2α 2 1, z U. D n f(z) Remark 1. Geometric interpretation: If we denote with p α the analytic and univalent functions with the properties p α (0) = 1, p α(0) > 0 and p α (U) = Ω(α), where Ω(α) = {w = u + i v : v 2 < 4αu + u 2, u > 0} (note that Ω(α) is the interior of a hyperbola in the right half-plane which is symmetric about the real axis and has vertex at the origin), then f SH n (α) if and only if Dn+1 f(z) p D n α (z), where the symbol denotes the subordination in U. We have p α (z) = (1 + 2α) 2α, b = b(α) = f(z) 1 + bz 1 + 4α 4α2 1 z (1 + 2α) 2 and the branch of the square root w is chosen so that Im w 0. If we consider p α (z) = 1 + C 1 z +..., we have C 1 = 1 + 4α 1 + 2α. Remark 2. If we denote by = G(z),we have: g SH n (α) if and only if G SH(α) = SH 0 (α). Theorem 1. (see [2]) If F (z) SH n (α), α > 0, n N, and f(z) = L a F (z), where L a is the integral operator defined by (1), then f(z) SH n (α), α > 0, n N. Definition 3. (see [1]) Let f A and α > 0. We say that the function f is in the class CCH(α) with respect to the function g SH(α) if zf (z) ( ) { } 2 g(z) 2α zf (z) ( ) 2 1 < Re + 2α 2 1, z U. g(z) Remark 3. Geometric interpretation: f CCH(α) with respect to the function g SH(α) if and only if zf (z) take all values in the convex domain Ω(α), g(z) where Ω(α) is defined in Remark 1.
26 Mugur Acu Theorem 2. (see [1]) If f(z) = z + a j z j belong to the class CCH(α), α > 0, with respect to the function g(z) SH(α), α > 0, g(z) = z + then a 2 1 + 4α 1 + 2α, a 3 (1 + 4α)(11 + 56α + 72α2 ) 12(1 + 2α) 3. b j z j, The next theorem is result of the so called admissible functions method due to P.T. Mocanu and S.S. Miller (see [3], [4], [5]). Theorem 3. Let q be convex in U and j : U C with Re[j(z)] > 0, z U. If p H(U) and satisfied p(z) + j(z) zp (z) q(z), then p(z) q(z). 3 Main results Definition 4. Let f A, n N and α > 0. We say that the function f is in the class CCH n (α), with respect to the function g SH n (α), if D n+1 f(z) ( ) { } 2 2α 2 1 D n+1 f(z) ( ) < Re +2α 2 1, z U. Remark 4. Geometric interpretation: f CCH n (α), with respect to the function g SH n (α), if and only if Dn+1 f(z) p D n α (z), where the symbol g(z) denotes the subordination in U and p α is defined in Remark 1. Remark 5. If we denote D n f(z) = F (z) and = G(z) we have: f CCH n (α), with respect to the function g SH n (α), if and only if F CCH(α), with respect to the function G SH(α) (see Remark 2). Theorem 4. Let α > 0, n N and f CCH n (α), f(z) = z +a 2 z 2 +a 3 z 3 +..., with respect to the function g SH n (α), then a 2 1 2 n 1 + 4α 1 + 2α, a 3 1 3 n (1 + 4α)(11 + 56α + 72α2 ) 12(1 + 2α) 3.
On a subclass of n-close to convex functions... 27 Proof. If we denote by D n f(z) = F (z), F (z) = b j z j, we have (using Remark 5) from the above series expansions we obtain a j 1 j n b j, j 2. Using the estimations from the Theorem 2 we obtain the needed results. Theorem 5. Let α > 0 and n N. If F (z) CCH n (α), with respect to the function G(z) SH n (α), and f(z) = L a F (z), g(z) = L a G(z), where L a is the integral operator defined by (1), then f(z) CCH n (α), with respect to the function g(z) SH n (α). Proof. By differentiating (1) we obtain (1 + a)f (z) = af(z) + zf (z) and (1 + a)g(z) = ag(z) + zg (z). or obtain Thus (2) By means of the application of the linear operator D n+1 we obtain (1 + a)d n+1 F (z) = ad n+1 f(z) + D n+1 (zf (z)) (1 + a)d n+1 F (z) = ad n+1 f(z) + D n+2 f(z) Similarly, by means of the application of the linear operator D n = h(z), by simple cal- With notations Dn+1 f(z) culations, we have (1 + a)d n G(z) = a + D n+1 g(z) D n+1 F (z) = Dn+2 f(z) + ad n+1 f(z) = D n G(z) D n+1 g(z) + a D n+2 f(z) D = n+1 g(z) Dn+1 g(z) + a Dn+1 f(z) D n+1 g(z) + a = p(z) and Dn+1 g(z) D n+2 f(z) D n+1 g(z) = p(z) + 1 h(z) zp (z) we
28 Mugur Acu Thus from (2) we obtain (3) D n+1 F (z) D n G(z) = h(z) ( ) zp 1 (z) h(z) + p(z) + a p(z) = h(z) + a 1 = p(z) + h(z) + a zp (z) From Remark 4 we have Dn+1 F (z) p D n α (z) and thus, using (3), we G(z) obtain 1 p(z) + h(z) + a zp (z) p α (z). 1 We have from Remark 1 and from the hypothesis Re h(z) + a > 0, z U. In this conditions from Theorem 3 we obtain p(z) p α (z)or Dn+1 f(z) p α(z). This means that f(z) = L a F (z) CCH n (α), with respect to the function g(z) = L a G(z) SH n (α) (see Theorem 1). Theorem 6. Let a C, Re a 0, α > 0, and n N. If F (z) CCH n (α), with respect to the function G(z) SH n (α), F (z) = z + a j z j, and g(z) = L a G(z), f(z) = L a F (z), f(z) = z + b j z j, where L a is the integral operator defined by (1), then b 2 a + 1 a + 2 1 2 1 + 4α n 1 + 2α, b 3 a + 1 a + 3 1 3 (1 + 4α)(11 + 56α + 72α2 ). n 12(1 + 2α) 3 Proof. From f(z) = L a F (z) we have (1 + a)f (z) = af(z) + zf (z). Using the above series expansions we obtain (1 + a)z + (1 + a)a j z j = az + ab j z j + z + jb j z j
On a subclass of n-close to convex functions... 29 and thus b j (a + j) = (1 + a)a j, j 2. From the above we have b j a + 1 a + j a j, j 2. Using the estimations from Theorem 4 we obtain the needed results. For a = 1, when the integral operator L a become the Libera integral operator, we obtain from the above theorem: Corollary 1. Let α > 0 and n N. If F (z) CCH n (α), with respect to the function G(z) SH n (α), F (z) = z + a j z j, and g(z) = L (G(z)), f(z) = L (F (z)), f(z) = z + defined by L (H(z)) = 2 z z 0 b j z j, where L is Libera integral operator H(t)dt, then b 2 1 1 + 4α 2n 1 3 + 6α, b 3 1 3 (1 + 4α)(11 + 56α + 72α2 ). n 24(1 + 2α) 3 Theorem 7. Let n N and α > 0. If f CCH n+1 (α) then f CCH n (α). Proof. With notations Dn+1 f(z) (see the proof of the Theorem 5): = p(z) and Dn+1 g(z) D n+2 f(z) D n+1 g(z) = p(z) + 1 h(z) zp (z). = h(z) we have From f CCH n+1 (α) we obtain (see Remark 4) p(z) + 1 h(z) zp (z) p α (z). Using the Remark 1 we have Re 1 > 0, z U, and from Theorem h(z) 3 we obtain p(z) p α (z) or f CCH n (α). Remark 6. From the above theorem we obtain CCH n (α) CCH 0 (α) = CCH(α) for all n N.
30 Mugur Acu References [1] M. Acu, Close to convex functions associated with some hyperbola, (to appear). [2] M. Acu, On a subclass of n-starlike functions associated with some hyperbola, (to appear). [3] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent functions, Mich. Math. 28 (1981), 157-171. [4] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet differential equations, J. Differential Equations 56 (1985), 297-308. [5] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Mich. Math. 32(1985), 185-195. [6] Gr. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013, Springer-Verlag, 1983, 362-372. [7] J. Stankiewicz, A. Wisniowska, Starlike functions associated with some hyperbola, Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka 19(1996), 117-126. Lucian Blaga University of Sibiu Department of Mathematics Str. Dr. I. Raṭiu, No. 5-7 550012 - Sibiu, Romania E-mail address: acu mugur@yahoo.com