International Journal of Analysis and Applications ISSN 2291-8639 Volume 12, Number 2 (2016), 157-162 http://www.etamaths.com APPLICATION OF HYPERGEOMETRIC DISTRIBUTION SERIES ON CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS TRAILOKYA PANIGRAHI∗ Abstract. The object of the present paper is to give some characterizations for hypergeometric distribution series to be in various subclasses of analytic functions. 1. Introduction Let A denote the family of all functions f analytic in U := {z ∈ C : |z| < 1} with the usual normalization condition f(0) = f′(0) − 1 = 0. Thus f has the following Taylor-Maclaurin series: (1) f(z) = z + ∞∑ l=2 alz l. Let S be the subclass of A consisting of all functions f of the form (1) which are univalent in U. A function f ∈A is said to be in k−UCV, the class of k-uniformly convex function (0 ≤ k < ∞) if f ∈S along with the property that for every circular arc γ contained in U with center ξ where |ξ| < k, the image curve f(γ) is a convex arc. It is well-known that [5] f ∈ k−UCV if and only if the image of the function p, where p(z) = 1 + zf′′(z) f′(z) (z ∈ U) is a subset of the conic region (2) Ωk = {w = u + iv : u2 > k2(u− 1)2 + k2v2, 0 ≤ k < ∞}. The class k−ST consisting of k-uniformly starlike functions is defined via k−UCV by the Alexander transform i.e. f ∈ k −ST ⇐⇒ g ∈ k −UCV where g(z) = ∫ z 0 f(t) t dt. The class k−ST and its properties were investigated in [6]. The analytic characterization of k−UCV and k −ST are given as below: (3) k −UCV = {f ∈A : < ( 1 + zf′′(z) f′(z) ) > k ∣∣∣∣zf′′(z)f′(z) ∣∣∣∣ (z ∈ U)} and (4) k −ST = {f ∈A : < ( zf′(z) f(z) ) > k ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ (z ∈ U)} Note that for k = 0 and k = 1, we get 0−UCV = K, 0−ST = S∗, 1−UCV = UCV and 1−ST = SP, where K, S∗, UCV, SP are respectively the familiar classes of univalent convex functions, univalent starlike functions [3], uniformly convex functions [4] (also, see [7, 12]) and parabolic starlike functions [12]. For two analytic functions f and g in U, the function f is said to be subordinate to g or g is said to be superordinate to f, if there exists a function w analytic in U with |w| ≤ |z| such that f(z) = g(w(z)). In such case, we write f ≺ g or f(z) ≺ g(z). If the function g is univalent in U, then f ≺ g ⇐⇒ f(0) = g(0) and f(U) ⊂ g(U) (see, for detail [8]). 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. analytic functions; k-uniformly convex functions; k-uniformly starlike functions; hypergeo- metric distribution series. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 157 158 PANIGRAHI Making use of subordination between analytic functions, Aouf [1] introduced and studied the class Rλ(A,B,α) as follows: Definition 1.(see[1, with p=1]) For −1 ≤ A < B ≤ 1, |λ| < π 2 and 0 ≤ α < 1, we say that a function f(z) ∈A is in the class Rλ(A,B,α) if it satisfies the following subordination condition: (5) eiλf′(z) ≺ cosλ [ (1 −α) 1 + Az 1 + Bz + α ] + isinλ. The subordination (5) is equivalent to the inequality (6) given below: (6) ∣∣∣∣ eiλ(f′(z) − 1)Beiλf′(z) − [Beiλ + (A−B)(1 −α)cosλ] ∣∣∣∣ < 1 (z ∈ U). For particular values of parameters A,B,α and λ, we obtain various subclasses of analytic functions studied by different researchers (for details, see [2]). In 1998, Ponnusamy and Ronning [10] introduced and studied the classes S∗β and Cβ consisting of functions of the form (1) satisfying the following conditions: (7) S∗β = { f ∈A : ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ < β (z ∈ U, β > 0)}, and (8) Cβ = { f ∈A : ∣∣∣∣zf′′(z)f′(z) ∣∣∣∣ < β (z ∈ U, β > 0)}. It is worthy to mention here that f ∈Cβ ⇐⇒ zf′ ∈S∗β (β > 0). Recently, we introduced a new series H(M,N,n; z) whose coefficient are probabilities of hypergeometric distribution as follows: (9) H(M,N,n; z) = z + 1( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) zl. Let us define the linear operator J(M,N,n) : A−→A given by (10) J(M,N,n)f(z) = H(M,N,n; z) ? f(z) = z + 1( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) alz l (z ∈ U), where ? denote the convolution or Hadamard product between two analytic functions. Motivated by the works of [9, 10, 13], in this paper we investigate some characterization for hyperge- ometric distribution series to be in the subclasses S∗β and Cβ of analytic functions. 2. Preliminaries lemmas To prove our main results, we need the following lemmas. Lemma 1. (see [1], Theorem 4 with p=1) A sufficient condition for f(z) defined by (1) to be in the class Rλ(A,B,α) is (11) ∞∑ l=2 l(1 + |B|)|al| ≤ (B −A)(1 −α)cosλ. Lemma 2. (see [6]) Let f(z) ∈A. If for some k, the following inequality (12) ∞∑ l=2 (l + k(l− 1))|al| ≤ 1 holds true, then f ∈ k −ST . Lemma 3. (see [5, 11]) A function f ∈A of the form (1) is in k −UCV if it satisfies the condition (13) ∞∑ l=2 l[l(k + 1) −k]|al| ≤ 1. APPLICATION OF HYPERGEOMETRIC DISTRIBUTION... 159 Another sufficient condition for the class k −UCV is given in [7] as follows: Lemma 4. (see [7, 11]) Let f ∈S be of the form (1). If for some k (0 ≤ k < ∞), the inequality (14) ∞∑ l=2 l(l− 1)|al| ≤ 1 k + 2 , holds true, then f ∈ k −UCV. The number 1 k+2 cannot be increased. Lemma 5. (see [11]) Let f ∈A be of the form (1). If the inequality (15) ∞∑ l=2 [β + l− 1]|al| ≤ β (β > 0), is satisfied, then f ∈S∗β. Lemma 6. (see [11]) Let f ∈A be of the form (1). If (16) ∞∑ l=2 l[β + l− 1]|al| ≤ β (β > 0), then f ∈ Cβ. Lemma 7. (see [1], Theorem 1 with p=1) Let the function f(z) defined by (1) be in the class Rλ(A,B,α), then (17) |al| ≤ (B −A)(1 −α)cosλ l (l ≥ 2). 3. Main Results Unless otherwise stated, we assume throughout the sequel that −1 ≤ A < B ≤ 1, |λ| < π 2 , 0 ≤ α < 1. Theorem 1. Let k ≥ 0. If the inequality (18) 1( N n ) [M(k + 1)A1 −(N −M n )] ≤ secλ (B −A)(1 −α) − 1, where (19) A1 = ∞∑ l=2 ( M − 1 l− 2 )( N −M n− l + 1 ) is satisfied, then J(M,N,n) maps the class Rλ(A,B,α) into k −UCV. Proof. Let the function f given by (1) be a member of Rλ(A,B,α). By (10), we have J(M,N,n)f(z) = z + 1( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) alz l. In view of Lemma 3, it is sufficient to show that 1( N n ) ∞∑ l=2 l[l(k + 1) −k] ( M l− 1 )( N −M n− l + 1 ) |al| ≤ 1. By making use of Lemma 7, it is again sufficient to prove that (20) P1 = 1( N n ) ∞∑ l=2 [l(k + 1) −k] ( M l− 1 )( N −M n− l + 1 ) ≤ secλ (B −A)(1 −α) . 160 PANIGRAHI Now P1 = 1( N n ) ∞∑ l=2 [(l− 1)(k + 1) + 1] ( M l− 1 )( N −M n− l + 1 ) = 1( N n ) [ ∞∑ l=2 (k + 1) M! (l− 2)!(M − l + 1)! ( N −M n− l + 1 ) + ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 )] = M(k + 1)( N n ) ∞∑ l=2 ( M − 1 l− 2 )( N −M n− l + 1 ) + 1( N n ) [ ∞∑ l=0 ( M l )( N −M n− l ) − ( N −M n )] = M(k + 1)( N n ) A1 − ( N−M n )( N n ) + 1, where A1 is defined as in (19). Thus, in view of (20), if the inequality (18) is satisfied, then J(M,N,n)(f) ∈ k −UCV as asserted. The proof of Theorem 1 is complete. � Theorem 2. If the inequality (21) M( N n )A1 ≤ secλ (k + 2)(B −A(1 −α) is satisfied, then J(M,N,n) maps the class Rλ(A,B,α) into k −UCV. Proof. Let the function f given by (1) be a member of Rλ(A,B,α). By virtue of Lemma 4, it is sufficient to show that 1( N n ) ∞∑ l=2 l(l− 1) ( M l− 1 )( N −M n− l + 1 ) |al| ≤ 1 k + 2 Using the coefficient estimate (17), it is again sufficient to show that (22) P2 = 1( N n ) ∞∑ l=2 (l− 1) ( M l− 1 )( N −M n− l + 1 ) ≤ secλ (k + 2)(B −A)(1 −α) . Now, P2 = M( N n ) ∞∑ l=2 ( M − 1 l− 2 )( N −M n− l + 1 ) = M( N n )A1. In view of (22), if the condition (21) is satisfied, then J(M,N,n)(f) ∈ k−UCV as asserted. This ends the proof of Theorem 2. � Theorem 3. If the inequality (23) (1 + k) − (1 + k)( N n ) (N −M n ) − k( N n ) (M + 1) B1 ≤ secλ (B −A)(1 −α) , where (24) B1 = ∞∑ l=2 ( M + 1 l )( N −M n− l + 1 ) is satisfied, then J(M,N,n) maps the class Rλ(A,B,α) into k −ST . Proof. Let the function f given by (1) be a member of Rλ(A,B,α). By virtue of Lemma 2, it is sufficient to show that 1( N n ) ∞∑ l=2 [l + k(l− 1)] ( M l− 1 )( N −M n− l + 1 ) |al| ≤ 1. Using the coefficient estimate (17), it is again sufficient to show that (25) P3 = 1( N n ) ∞∑ l=2 [l + k(l− 1)] l ( M l− 1 )( N −M n− l + 1 ) ≤ secλ (B −A)(1 −α) APPLICATION OF HYPERGEOMETRIC DISTRIBUTION... 161 Now, P3 = 1( N n ) ∞∑ l=2 [ 1 + (1 − 1 l )k ]( M l− 1 )( N −M n− l + 1 ) = 1( N n ) ∞∑ l=2 [ (1 + k) − k l ]( M l− 1 )( N −M n− l + 1 ) = (1 + k) [ 1 − ( N−M n )( N n ) ] − k (M + 1) ( N n )B1. Therefore, in view of (25), if the inequality (23) is satisfied, then J(M,N,n)(f) ∈ k−ST as asserted. This complete the proof of Theorem 3. � Theorem 4. If f ∈Rλ(A,B,α) and the inequality (26) 1 − ( N−M n )( N n ) ≤ 1 1 + |B| , is satisfied, then J(M,N,n)(f) ∈Rλ(A,B,α). Proof. Let the function f ∈ A given by (1) be a member of Rλ(A,B,α). By virtue of Lemma 1 and the coefficient inequality (17) it is sufficient to show that (27) P4 = 1( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) ≤ 1 1 + |B| . Now P4 is equivalently written as P4 = ∞∑ l=1 ( M l )( N−M n−l )( N n ) = 1 − (N−Mn )( N n ) Thus, in view of (27), if the inequality (26) is satisfied, then J(M,N,n)(f) ∈Rλ(A,B,α). The proof of Theorem 4 is complete. � Theorem 5. Let β > 0, f ∈Rλ(A,B,α) and the inequality (28) β − 1 (M + 1) ( N n )B1 − ( N−M n )( N n ) ≤ βsecλ (B −A)(1 −α) − 1, is satisfied, then J(M,N,n)(f) ∈S∗β . Proof. By making use of Lemma 5, it is sufficient to show that ∞∑ l=2 (β + l− 1) ( M l−1 )( N−M n−l+1 )( N n ) |al| ≤ β. Since f ∈Rλ(A,B,α), using the coefficient estimate (17), it is sufficient to show that (29) P5 = 1( N n ) ∞∑ l=2 [ β + l− 1 l ]( M l− 1 )( N −M n− l + 1 ) ≤ βsecλ (B −A)(1 −α) . Now, P5 = 1( N n ) ∞∑ l=2 ( β − 1 l )( M l− 1 )( N −M n− l + 1 ) + 1( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) = β − 1 (M + 1) ( N n )B1 − ( N−M n )( N n ) + 1. Thus, in view of (29), if the inequality (28) is satisfied, then J(M,N,n)(f) ∈ S∗β as asserted. This proof the Theorem 5. � 162 PANIGRAHI Theorem 6. Let β > 0. If the inequality (30) 1( N n ) [MA1 −β(N −M n )] ≤ β [ secλ (B −A)(1 −α) − 1 ] is satisfied, then J(M,N,n) maps the class Rλ(A,B,α) into Cβ. Proof. In view of Lemma 6, it is sufficient to show that 1( N n ) ∞∑ l=2 l[β + l− 1] ( M l− 1 )( N −M n− l + 1 ) |al| ≤ β. Using coefficient inequality (17), it is enough to show that (31) P6 = 1( N n ) ∞∑ l=2 [β + l− 1] ( M l− 1 )( N −M n− l + 1 ) ≤ βsecλ (B −A)(1 −α) . Now the expression P4 can be equivalently written as P6 = β( N n ) ∞∑ l=2 ( M l− 1 )( N −M n− l + 1 ) + ∞∑ l=2 M! (l− 2)!(M − l + 1)! ( N −M n− l + 1 ) = β −β ( N−n n )( N n ) + M( N n ) ∞∑ l=2 ( M − 1 l− 2 )( N −M n− l + 1 ) = M( N n )A1 − β ( N−M n )( N n ) + β. Thus, in view of (31) if the inequality (30) is satisfied, then J(M,N,n)(f) ∈Cβ as desired. The proof of Theorem 6 is thus completed. � References [1] M. K. Aouf, On certain subclass of analytic p-valent functions of order alpha, Rend. Mat., 7(8) (1988), 89-104. [2] M. K. Aouf, A. O. Mostafa and H. M. 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