1_xxx_srivastava.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 3, 2009, (302-324) ISSN 1307-5543 – www.ejpam.com A Unified Class of Analytic Functions with Varying Argument of Coefficients J. Dziok1 and H.M. Srivastava2∗ 1 Institute of Mathematics, University of Rzeszów, ul. Rejtana 16A, PL-35-310 Rzeszów, Poland 2 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada Abstract. The object of the present paper is to investigate several classes of analytic functions with varying argument of coefficients, which are defined here by means of the principle of sub- ordination between analytic functions. Such properties as the coefficient estimates, distortion theorems, subordination theorems, convolution properties, integral means inequalities, and radii of conexity and starlikenes are investigated. Some consequences of our main results for new or known classes of analytic functions are also pointed out. 2000 Mathematics Subject Classifications: Primary 30C45, 30C50; Secondary 30C55. Key Words and Phrases: Analytic functions; Convex functions; k-Starlike functions; Uni- formly convex functions; Varying arguments of coefficients; Subordination between analytic functions; Hadamard product (or convolution); Generalized hypergeometric function; Dziok- Srivastava operator. ∗Corresponding author. Email addresses: jdziok�univ.rzeszow.pl (J. Dziok), harimsri�math.uvi . a (H. Srivastava) http://www.ejpam.com 302 c© 2009 EJPAM All rights reserved. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 303 1. Introduction and Definitions LetB denote the class of functions f : U−→ C, where U := U(1) and U(r) := {z : z ∈ C and |z|< r}. We also denote by fA the class of functions f ∈ B which are analytic in U (see, for details, [32]). We say that a function f ∈B is subordinate to a function F ∈B , and we write f (z)≺ F(z) or, simply, f ≺ F, if and only if there exists a function w ∈B with |w(z)| ≦ |z| (z ∈ U) , such that (see, for details, [15]) f (z) = F � w(z) � (z ∈ U) . In particular, if F is univalent in U, we have the following equivalence: f (z) ≺ F(z)⇐⇒ f (0) = F(0) and f (U) ⊂ F(U). For functions f , g ∈ fA of the forms: f (z) = ∞∑ n=0 anzn and g(z) = ∞∑ n=0 bnzn, by f ∗ g we denote the Hadamard product (or convolution) of f and g, defined by � f ∗ g � (z) := ∞∑ n=0 an bnzn =: � g ∗ f � (z) (z ∈ U) . J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 304 We denote byA the class of functions f ∈B of the form: f (z) = z+ ∞∑ n=2 anzn (z ∈ U). (1.1) We also denote by Tη � η ∈ R � the class of functions f ∈ A of the form (1.1) for which arg(an) = π+ (1− n)η (n ∈ N \ {1}; N := {1, 2, 3, · · · }). (1.2) For η = 0, we obtain the familiar class T0 of functions with negative coefficients. Moreover, we define T := ⋃ η∈R Tη. (1.3) The class T was introduced by Silverman [24] (see also [31]). It is called the class of functions with varying argument of coefficients. Let α ∈ [0, 1) and r ∈ (0, 1]. A function f ∈A is said to be convex of order α in U(r) if and only if ℜ � 1+ z f ′′(z) f ′(z) � > α � z ∈ U(r); 0 ≦ α < 1; 0< r ≦ 1 � . A function f ∈A is said to be starlike of order α in U(r) if and only if ℜ � z f ′(z) f (z) � > α � z ∈ U(r); 0≦ α < 1; 0 < r ≦ 1 � . (1.4) We denote by S c (α) the class of all functions f ∈ A , which are convex of order α in U, and by S ∗ (α) we denote the class of all functions f ∈ A , which are starlike of order α in U. We also set S c = S c(0) and S ∗ = S ∗(0). J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 305 It is easy to show that, for a function f ∈ Tη, the condition (1.4) is equivalent to the following inequality: ���� z f ′(z) f (z) − 1 ����< 1−α � z ∈ U(r); 0≦ α < 1; 0< r ≦ 1 � . (1.5) LetB be a subclass of the class A . We define the radius of starlikeness of order α and the radius of convexity of order α for the classB by R∗ α (B) = inf f ∈B � sup � r ∈ (0, 1] : f is starlike of order α in U(r) � and Rc α (B) = inf f ∈B � sup � r ∈ (0, 1] : f is convex of order α in U(r) � , respectively. Let k, A and B be real parameters such that k ≧ 0, 0 ≦ B ≦ 1 and − 1 ≦ A< B. Also let ϕ,φ ∈ A . By W � φ,ϕ; A, B; k � we denote the class of functions f ∈ A such that � ϕ ∗ f � (z) 6= 0 (z ∈ U \ {0}) and � φ ∗ f � (z)� ϕ ∗ f � (z) − k ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ���� ≺ 1+ Az 1+ Bz . (1.6) If 0≦ B < 1, then the condition (1.6) is equivalent to the following inequality: ����� � φ ∗ f � (z)� ϕ ∗ f � (z) − k ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ����− 1− AB 1− B2 �����< B− A 1− B2 (z ∈ U). (1.7) On the other hand, if B = 1, then we have ℜ �� φ ∗ f � (z)� ϕ ∗ f � (z) � − k ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ����> 1+ A 2 (z ∈ U) . (1.8) J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 306 Related to the function classes T and Tη, we define the following two classes: T W � φ,ϕ; A, B; k � := T ∩W � φ,ϕ; A, B; k � and T W η � φ,ϕ; A, B; k � := Tη ∩W � φ,ϕ; A, B; k � . For our present investigation, we assume that ϕ and φ are the functions of the following forms: ϕ(z) = z + ∞∑ n=2 αnzn and φ(z) = z + ∞∑ n=2 β nzn (z ∈ U), (1.9) where the real sequences � αn and � β n are constrained further by 0≦ αn < β n (n ∈ N \ {1}) . Moreover, let us put dn := (k+ 1) (1+ B)β n− (kB + A+ k+ 1)αn (n ∈ N \ {1}) . (1.10) The function classesW � φ,ϕ; A, B; k � andWη � φ,ϕ; A, B; k � unify and extend var- ious known classes of analytic functions. We choose to list a few of these associated analytic function classes in the last section (Section 8). The object of the present paper is to investigate coefficient estimates, distortion theorems, subordination theorems, convolution properties, integral means inequali- ties, and radii of conexity and starlikenes of functions in the general classes T W η � φ,ϕ; A, B; k � and T W � φ,ϕ; A, B; k � , which we have introduced here. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 307 2. Coefficient Inequalities and Coefficient Estimates In this section, we first derive a sufficient condition for a function f to belong to the class W � φ,ϕ; A, B; k � . Theorem 2.1. Let � dn be defined by (1.10), 0 ≦ B ≦ 1, and −1 ≦ A < B. If a function f of the form (1.1) with � ϕ ∗ f � (z) 6= 0 (z ∈ U \ {0}), satisfies the following condition: ∞∑ n=2 dn ��an �� ≦ B − A, (2.1) then the function f belongs to the class W � φ,ϕ; A, B; k � . Proof. Let 0 ≦ B < 1. Then, for a function f of the form (1.1), we have ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − k ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ����− 1− AB 1− B2 ���� ≦ (k+ 1) ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ����+ B (B − A) 1− B2 ≦ (k+ 1) ∞∑ n=2 � βn−αn � |an||z| n−1 1− ∞∑ n=2 αn|an||z|n−1 + B (B− A) 1− B2 . Thus, by (2.1), we obtain (1.7). Consequently, f ∈ W � φ,ϕ; A, B; k � . We now sup- pose that B = 1. Then simple calculations give k ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ����−ℜ �� φ ∗ f � (z)� ϕ ∗ f � (z) − 1+ A 2 � ≦ (k+ 1) ���� � φ ∗ f � (z)� ϕ ∗ f � (z) − 1 ���� J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 308 ≦ (k+ 1) ∞∑ n=2 � β n−αn � |an||z| n−1 1− ∞∑ n=2 αn|an||z|n−1 , which, by means of (2.1), leads us to (1.8). Therefore, f ∈ W � φ,ϕ; A, B; k � , and the proof of Theorem 2.1 is completed. Our next theorem shows that the condition (2.1) is necessary as well for func- tions of the form (1.1) satisfying the argument property (1.2) to belong to the class T W η � φ,ϕ; A, B; k � . Theorem 2.2. Let f be a function of the form (1.1) satisfying the argument property (1.2). Then f belongs to the class T W η � φ,ϕ; A, B; k � if and only if the condition (2.1) holds true. Proof. In view of Theorem 2.1, we need only to show that each function f from the class T W η � φ,ϕ; A, B; k � satisfies the coefficient inequality (2.1). Let f be a function of the form (1.1) and satisfying the argument property (1.2) belong to the class T W η � φ,ϕ; A, B; k � . Then, putting z = r iη in the conditions (1.7) and (1.8), we obtain (k+ 1) ∞∑ n=2 � β n−αn � |an|r n−1 1− ∞∑ n=2 αn|an|rn−1 < B− A 1+ B . We thus find that ∞∑ n=2 � (k+ 1) (1+ B)β n− (k (1+ B) + 1+ A)αn � |an|r n−1 < B − A, which, upon letting r → 1−, readily yields the assertion (2.1). Since the condition (2.1) is independent of η, Theorem 2.2 yields the following result. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 309 Theorem 2.3. Let f be a function of the form (1.1) satisfying the agument property (1.2). Then f ∈ T W � φ,ϕ; A, B; k � if and only if the condition (2.1) holds true. From Theorems 2.2 and 2.3 we can obtain the following coefficient estimates for functions in the classes T W η � φ,ϕ; A, B; k � and T W � φ,ϕ; A, B; k � , respectively. Corollary 2.1. If a function f of the form (1.1) belongs to the class T W η � φ,ϕ; A, B; k � , then ��an ��≦ B− A dn (n ∈ N \ {1}), (2.2) where dn is defined by (1.10). The result is sharp and the functions fn,η given by fn,η(z) = z− B − A dn ei(1−n)ηzn (z ∈ U; n ∈ N) (2.3) are the extremal functions. Corollary 2.2. If a function f of the form (1.1) belongs to the class T W � φ,ϕ; A, B; k � , then the coefficient estimates given by (2.2) hold true. The result is sharp and the func- tions fn,η � η ∈ R � given by (2.3) are the extremal functions. 3. Distortion Theorems By applying Theorem 2.2, we can deduce the following lemma. Lemma 3.1. Let a function f of the form (1.1) belong to the class T W η � φ,ϕ; A, B; k � . If the sequence � dn defined by (1.10) satisfies the following inequality: d2 ≦ dn (n ∈ N \ {1}) , (3.1) J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 310 then ∞∑ n=2 an ≦ B− A d2 . Moreover, if nd2 ≦ 2dn (n ∈ N \ {1}) , (3.2) then ∞∑ n=2 nan ≦ 2 (B− A) d2 . Theorem 3.1. Let a function f belong to the class T W η � φ,ϕ; A, B; k � . If the sequence � dn defined by (1.10) satisfies (3.1), then r − B− A d2 r2 ≦ �� f (z) �� ≦ r + B− A d2 r2 (|z|= r < 1) . (3.3) Moreover, if (3.2) holds true, then 1− 2 (B− A) d2 r ≦ �� f ′(z) ��≦ 1+ 2 (B− A) d2 r (|z|= r < 1) . (3.4) The result is sharp and the extremal function f2,η is given by (2.3). Proof. Let a function f of the form (1.1) belong to the class T W η � φ,ϕ; A, B; k � . Then, for |z|= r < 1, �� f (z) �� = �����z + ∞∑ n=2 anzn ����� ≦ r + ∞∑ n=2 ��an �� rn = r + r2 ∞∑ n=2 ��an �� rn−2 ≦ r + r2 ∞∑ n=2 ��an �� and �� f (z) �� = �����z + ∞∑ n=2 anzn ����� ≧ r − ∞∑ n=2 ��an �� rn = r − r2 ∞∑ n=2 ��an �� rn−2 ≧ r−r2 ∞∑ n=2 ��an �� , which, in light of Lemma 3.1, yields (3.3). Analogously, we can prove (3.4). Theorem 3.1 implies the following corollary: J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 311 Corollary 3.1. Let a function f belong to the class T W � φ,ϕ; A, B; k � . If the sequence � dn defined by (1.10) satisfies (3.1), then the assertion (3.3) holds true. Moreover, if we assume that (3.2) is satisfied, then the assertion (3.4) holds true. The result is sharp and the extremal functions f2,η � η ∈ R � are given by (2.3). 4. Results Involving Subordination Between Analytic Functions Before stating and proving our subordination theorems for the function classes T W η � φ,ϕ; A, B; k � and T W � φ,ϕ; A, B; k � , we need the following definition as well as Lemma 4.1. Definition 4.1. A sequence {bn} of complex numbers is said to be a subordinating factor sequence if, for each function f of the form (1.1) from the class S c, we have ∞∑ n=1 bnanzn ≺ f (z) � a1 = 1 � . (4.1) Lemma 4.1. (see [36]) The sequence {bn} is a subordinating factor sequence if and only if ℜ 1+ 2 ∞∑ n=1 bnzn ! > 0 (z ∈ U) . (4.2) Theorem 4.1. Let the sequence � dn , defined by (1.10), satisfy the inequality (3.1). If g ∈ S c and f ∈ T W η � φ,ϕ; A, B; k � , then ǫ( f ∗ g)(z)≺ g(z) (4.3) and ℜ � f (z) � >− 1 2ǫ (z ∈ U) , (4.4) where ǫ = d2 2 � B− A+ d2 � . (4.5) The constant factor ǫ cannot be replaced by a larger number. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 312 Proof. Let a function f of the form (1.1) belong to the class T W η � φ,ϕ; A, B; k � and suppose that g(z) = z+ ∞∑ n=2 cnzn (z ∈ U) belongs to the class S c. Then ǫ( f ∗ g)(z) = ǫz + ∞∑ n=2 � ǫan � cnzn. Thus, by the above Definition, the subordination result (4.3) holds true if � ǫan ∞ n=1 � a1 = 1 � is a subordinating factor sequence. In view of Lemma 4.1, this is equivalent to the following inequality: ℜ 1+ 2 ∞∑ n=1 ǫanzn ! > 0 (z ∈ U) . (4.6) By (3.1) for |z|= r < 1, we have ℜ 1+ 2 ∞∑ n=1 ǫanzn ! = ℜ 1+ 2ǫz + ∞∑ n=2 d2 B − A+ d2 anzn ! ≧ 1− 2ǫr − r B− A+ d2 ∞∑ n=2 dn ��an �� rn−1. Consequently, by using Theorem 2.2, we obtain ℜ 1+ 2 ∞∑ n=1 ǫanzn ! ≧ 1− d2 B− A+ d2 r − B− A B − A+ d2 r > 0. This evidently proves the inequality (4.6) and hence the subordination result (4.3). The inequality (4.4) follows from (4.3) by taking g(z) = z 1− z = z+ ∞∑ n=2 zn (z ∈ U) . J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 313 We next observe that the function f2,η of the form (2.3) belongs to the class T W η � φ,ϕ; A, B; k � . It is easily verified that min � ℜ � ǫ f2,η (z) � = − 1 2 (z ∈ U) . This shows that the constant (4.5) cannot be replaced by any larger number. Directly from Theorem 4.1, we can obtain Theorem 4.2 below. Theorem 4.2. Let the sequence � dn , defined by (1.10), satisfy the inequality (3.1). If g ∈ S c and f ∈ T W � φ,ϕ; A, B; k � , then the conditions (4.3) and (4.4) hold true. The constant factor ǫ in (4.3) cannot be replaced by a larger number. 5. Integral Means Inequalities Following the work of Littlewood [11], we obtain here some integral means in- equalities for functions belonging to the class T W η � φ,ϕ; A, B; k � . Lemma 5.1. (see [11]) Let f , g ∈ fA . If f ≺ g, then ∫ 2π 0 �� f (reiθ ) ��η dθ ≦ ∫ 2π 0 ��g(reiθ ) ��η dθ � 0< r < 1; η > 0 � . (5.1) Silverman [23] found that the following function: g(z) = z − z2 2 (z ∈ U) , is often extremal over the family of functions with negative coefficients. He applied this function to resolve a certain integral means inequality, which was conjectured in [25] and settled in [26], that (5.1) holds true for all functions f with negative coefficients. Silverman [26] also proved his conjecture for some subclasses of T . Applying Lemma 5.1 and Theorem 2.2, we now prove the following result. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 314 Theorem 5.1. Let the sequence � dn defined by (1.10) satisfy the inequality (3.1). If f ∈ T W η � φ,ϕ; A, B; k � , then ∫ 2π 0 �� f (reiθ ) ��λ dθ ≦ ∫ 2π 0 �� f2,η(reiθ ) ��λ dθ (0 < r < 1; λ > 0) , (5.2) where f2,η(z) is defined by (2.3). Proof. For a function f of the form (1.1), the inequality (5.2) is equivalent to the following inequality: ∫ 2π 0 �����1+ ∞∑ n=2 anzn−1 ����� λ dθ ≦ ∫ 2π 0 ����1− B− A d2 e−iη z ���� λ dθ � z = reiθ � . Thus, by Lemma 5.1, it suffices to show that ∞∑ n=2 anzn−1 ≺− B − A d2 e−iηz. (5.3) Upon setting w(z) = ∞∑ n=2 d2eiη A− B an zn−1 (z ∈ U) , and using (3.1) and Theorem 2.2, we obtain |w(z)| = ����� ∞∑ n=2 d2 A− B an zn−1 �����≦ |z| ∞∑ n=2 dn B− A ��an ��≦ |z| (z ∈ U) . Since ∞∑ n=2 anzn−1 =− B − A d2 e−iηw(z) (z ∈ U) , by the definition of subordination, we have (5.3). This completes the proof of Theo- rem 5.1. We can restate Theorem 5.1 as Theorem 5.2 below. Theorem 5.2. Let the sequence � dn defined by (1.10) satisfy the inequality (3.1). If a function f of the form (1.1) satisfying the argument property (1.2) belongs to the class T W � φ,ϕ; A, B; k � , then the integral means inequality (5.2) holds true. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 315 6. The Radii of Convexity and Starlikeness We begin this section by proving the following result. Theorem 6.1. The radius of starlikeness of order α for the class T W η � φ,ϕ; A, B; k � is given by R∗ α � T W η � φ,ϕ; A, B; k �� = inf n∈N\{1} � (1−α) dn (n−α) (B − A) � 1 n−1 , (6.1) where dn is defined by (1.10). Proof. A function f ∈ Tη of the form (1.1) is starlike of order α in the disk U(r) (0< r ≦ 1) if and only if it satisfies the condition (1.5). Since ���� z f ′(z) f (z) − 1 ���� = �������� ∞∑ n=2 (n− 1 )anzn z + ∞∑ n=2 anzn �������� ≦ ∞∑ n=2 (n− 1 ) ��an �� |z|n−1 1− ∞∑ n=2 ��an �� |z|n−1 , by putting |z|= r, the condition (1.5) is true if ∞∑ n=2 n−α 1−α ��an �� rn−1 ≦ 1. (6.2) By Theorem 2.2, we have ∞∑ n=2 dn B− A ��an �� ≦ 1, so that the condition (6.2) is true if n−α 1−α rn−1 ≦ dn B− A (n ∈ N \ {1}), that is, if r ≦ � (1−α) dn (n−α) (B− A) � 1 n−1 (n ∈ N \ {1}). It follows that each function f ∈ T W η � φ,ϕ; A, B; k � is starlike of order α in the disk U(r), where r = R∗ α � T W η � φ,ϕ; A, B; k �� is defined by (6.1). J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 316 Theorem 6.2. The radius of convexity of order α for the class T W η � φ,ϕ; A, B; k � is given by Rc α � T W η � φ,ϕ; A, B; k �� = inf n∈N\{1} � (1−α) dn n (n−α) (B− A) � 1 n−1 , where dn is defined by (1.10). Proof. The proof of Theorem 6.2 is analogous to that of Theorem 6.1, and we omit the details involved. By applying Theorems 6.1 and 6.2, we obtain the following two corollaries. Corollary 6.1. The radius of starlikeness of order α for the class T W � φ,ϕ; A, B; k � is given by R∗ α � T W � φ,ϕ; A, B; k �� = inf n∈N\{1} � (1−α) dn (n−α) (B − A) � 1 n−1 , where dn is defined by (1.10). Corollary 6.2. The radius of convexity of order α for the class T W � φ,ϕ; A, B; k � is given by Rc α � T W � φ,ϕ; A, B; k �� = inf n∈N\{1} � (1−α) dn n (n−α) (B− A) � 1 n−1 , where dn is defined by (1.10). 7. Convolution Properties Theorem 7.1. Let the sequence � dn defined by (1.10) satisfy the inequality (3.1) with d2 ≧ B− A. If f ∈ T W η � φ,ϕ; A, B; k � and g ∈ T W µ � φ,ϕ; A, B; k � , then f ∗ g ∈ T W η+µ � φ,ϕ; A, B; k � . J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 317 Proof. Let the functions f and g of the forms: f (z) = z + ∞∑ n=2 anzn and g(z) = z+ ∞∑ n=2 bnzn (z ∈ U) . (7.1) belong to the classes T W η � φ,ϕ; A, B; k � and T W µ � φ,ϕ; A, B; k � , respectively. Then, by appealing to Theorem 2.2, we have ∞∑ n=2 dn B − A ��an �� ≦ 1 and ∞∑ n=2 dn B− A ��bn ��≦ 1. Thus, by the Cauchy-Schwarz inequality, we obtain ∞∑ n=2 dn B− A Æ��an bn ��≦ 1 . (7.2) In order to prove that ∞∑ k=2 dn B− A ��an bn ��≦ 1, by virtue of (7.2), it is sufficient to show that ��an bn ��≦ Æ��anbn �� (n ∈ N \ {1}) or, equivalently, that Æ��an bn �� ≦ 1 (n ∈ N \ {1}). We note from (7.2) that Æ��an bn �� ≦ B − A dn (n ∈ N \ {1}). Consequently, we need only to prove that B− A dn ≦ 1 (n ∈ N \ {1}). Since d2 ≧ B − A, by hypothesis, so the last inequality follows from (3.1). By applying Theorem 7.1, we obtain the following corollary. J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 318 Corollary 7.1. Let the sequence � dn defined by (1.10) satisfy the inequality (3.1) with d2 ≧ B− A. If f , g ∈ T W � φ,ϕ; A, B; k � , then f ∗ g ∈ T W � φ,ϕ; A, B; k � . Theorem 7.2. Let the sequence � dn defined by (1.10) satisfy the inequality (3.1) with d2 ≧ 2 (B− A) . If the functions f and g of the form (7.1) belong to the class T W η � φ,ϕ; A, B; k � , then the function h(z) given by h(z) = z− ∞∑ n=2 ( ��an ��2+ ��bn ��2) zn (7.3) belongs to the class T W 0 � φ,ϕ; A, B; k � . Proof. Suppose that each of the functions f and g of the form (7.1) belongs to the class T W η � φ,ϕ; A, B; k � . Then, by Theorem 2.2, we have ∞∑ n=2 � dn B − A ��an �� �2 ≦ 1 and ∞∑ n=2 � dn B− A ��bn �� �2 ≦ 1. We thus obtain ∞∑ n=2 1 2 � dn B− A �2���an ��2 + ��bn ��2 � ≦ 1. (7.4) In order to prove that ∞∑ k=2 dn B − A ���an ��2+ ��bn ��2 � ≦ 1, by means of (7.4), it is sufficient to show that dn B− A ≧ 2 (n ∈ N \ {1}). Since d2 ≧ 2 (B− A) , by hypothesis, so the last inequality follows from (3.1). J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 319 8. Concluding Remarks and Observations We conclude this paper by observing that, in view of the subordination relation (1.6), by suitably choosing the functions φ and ϕ, we can consider various new or known classes of functions. Let Wn � ϕ; A, B; k � :=W zϕ′ (z) , n−1∑ k=0 ϕ � x kz � ; A, B; k ! (n ∈ N; x = e 2πi n ). In particular, the class Wn � ϕ; A, B � :=Wn � ϕ; A, B; 0 � consists of functions f ∈A , which satisfy the following subordination condition: z � ϕ ∗ f �′ (z) n−1∑ k=0 � ϕ ∗ f �� x kz � ≺ 1+ Az 1+ Bz . It is related to the class of starlike functions with respect to n-symmetric points. More- over, by putting n = 1, we obtain the class W � ϕ; A, B � :=W1 � ϕ; A, B � defined by the following subordination condition: z � ϕ ∗ f �′ (z)� ϕ ∗ f � (z) ≺ 1+ Az 1+ Bz . This class is related to the familiar class of starlike functions. Analogously, the class defined by Hn � ϕ;γ, k � :=Wn � ϕ; 2γ− 1, 1; k � � 0 ≦ γ < 1 � consists of functions f ∈A , which satisfy the following condition: ℜ   z � ϕ ∗ f �′ (z) n−1∑ k=0 � ϕ ∗ f �� x kz � − γ   > k ��������� z � ϕ ∗ f �′ (z) n−1∑ k=0 � ϕ ∗ f �� x kz � − 1 ��������� (z ∈ U) . J. Dziok and H. Srivastava / Eur. J. Pure Appl. Math, 2 (2009), (302-324) 320 It is related to the class of k-uniformly convex functions of order γ with respect to n-symmetric points. Moreover, by setting n = 1, we obtain the function class H � ϕ;γ, k � :=H1 � ϕ;γ, k � , which is defined by the following condition: ℜ � z � ϕ ∗ f �′ (z)� ϕ ∗ f � (z) − γ � > k ����� z � ϕ ∗ f �′ (z)� ϕ ∗ f � (z) − 1 ����� (z ∈ U) . This class is related to the class of k-uniformly convex functions of order γ. The function classes UST � γ, k � :=H � z 1− z ;γ, k � and UCV � γ, k � :=H � z (1− z) 2 ;γ, k � are the well-known classes of of k-starlike functions of order γ and k-uniformly convex functions of order γ, respectively. In particular, the function classes UCV := UCV (1, 0) and k− UCV := UCV (k, 0) were introduced by Goodman [8] (see also [14] and [22]) and Kanas et al. ( [10] and [9]), respectively (see also [7], [20], [21], [33], [34] and [35]). We note that the following function class: HT � ϕ;γ, k � := T0 ∩H � ϕ;γ, k � was investigated recently by Raina and Bansal [19]. If we set h(α1, z) := z q Fs(α1, · · · ,αq;β1, · · · ,β s; z), where qFs is the generalized hypergeometric function (see, for details, [29]), then we obtain the following function class: UH � q, s,λ,γ, k � :=HT � λh(α1+ 1, z) + (1−λ)h(α1, z);γ, k � (0≦ λ ≦ 1) REFERENCES 321 which was introduced and studied by Ramachandran et al. [20] (see also several re- cent works including, for example, [4], [5], [6], [12], [13], [18], [27], [34] and [35], which investigate various properties and applications of the Dziok-Srivastava operator defined by means of the Hadamard product involving the generalized hyper- geometric function q Fs). Let λ be a convex parameter. A function f ∈A is said to belong to the class Vλ � ϕ; A, B � :=W � λ ϕ (z) z + (1−λ)ϕ′ (z) , z; A, B; 0 � if it satisfies the following condition: λ � ϕ ∗ f � (z) z + (1−λ) � ϕ ∗ f �′ (z)≺ 1+ Az 1+ Bz . Moreover, a function f ∈A is said to belong to the class Uλ � ϕ; A, B � :=W � λ ϕ (z) z + (1−λ)ϕ′ (z) ; A, B; 0 � if it satisfies the following condition: z � ϕ ∗ f �′ (z) + (1−λ) z2 � ϕ ∗ f �′′ (z) λ � ϕ ∗ f � (z) + (1− λ) z � ϕ ∗ f �′ (z) ≺ 1+ Az 1+ Bz . (8.1) The above-defined function classes Wn � ϕ; A, B � , Hn � ϕ;γ, k � , Uλ � ϕ; A, B � and Vλ � ϕ; A, B � generalize several important classes, many of which were investigated systematically in earlier works (see, for example, [1], [2], [3], [16], [17], [28] and [30]). If we apply the results presented in this paper to the classes discussed above, we can easily be led to a remarkably large number of additional new or known results. References [1] M. K. Aouf and H. M Srivastava, Some families of starlike functions with negative coef- ficients, J. Math. Anal. Appl. 203 (1996), 762–790. REFERENCES 322 [2] N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions de- fined by a class of multiplier transformations, Math. Comput. Modelling 37 (2003), 39– 49. [3] J. Dziok, On some applications of the Briot-Bouquet differential subordination, J. Math. Anal. Appl. 328 (2007), 295–301. [4] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the gener- alized hypergeometric function, Appl. Math. Comput. 103 (1999), 1–13. [5] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003), 7–18. [6] J. Dziok and H. M. Srivastava, Some subclasses of analytic functions with fixed argu- ment of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2002), 115–125. [7] A. Gangadharan, T. N. Shanmugam and H. M. Srivastava, Generalized hypergeomet- ric functions associated with k-uniformly convex functions, Comput. Math. Appl. 44 (2002), 1515–1526. [8] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92. [9] S. Kanas and H. M. Srivastava, Linear operators associated with k-uniformaly convex functions, Intergral Transform. Spec. Funct. 9 (2000), 121–132. [10] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327–336. [11] J. E. Littlewood, On inequalities in theory of functions, Proc. London Math. Soc. (Ser. 2) 23 (1925), 481–519. [12] J.-L. Liu and H. M. Srivastava, Certain properties of the Dziok-Srivastava operator, Appl. Math. Comput. 159 (2004), 485–493. [13] J.-L. Liu and H. M. Srivastava, A class of multivalently analytic functions associated with the Dziok-Srivastava operator, Integral Transform. Spec. Funct. 20 (2009), 401–417. [14] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165– 175. REFERENCES 323 [15] S. S. Miller and P. T. Mocanu, Differential Subordination: Theory and Applications, Se- ries on Monographs and Textbooks in Pure and Applied Mathematics, No. 225, Marcel Dekker Incorporated, New York and Basel, 2000. [16] J. Patel and A. K. Mishra, On certain subclasses of multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl. 332 (2007), 109– 122. [17] J. Patel, A. K. Mishra and H. M. Srivastava, Classes of multivalent analytic functions involving the Dziok–Srivastava operator, Comput. Math. Appl. 54 (2007), 599–616. [18] K. Piejko and J. Sokól, On the Dziok-Srivastava operator under multivalent analytic functions, Appl. Math. Comput. 177 (2006), 839–843. [19] R. K. Raina and D. Bansal, Some properties of a new class of analytic functions defined in terms of a Hadamard product, J. Inequal. Pure Appl. Math. 9 (1) (2008), Article 22, 1–9 (electronic). [20] C. Ramachandran, T. N. Shanmugam, H. M. Srivastava and A. Swaminathan, A unified class of k-uniformly convex functions defined by the Dziok-Srivastava linear operator, Appl. Math. Comput. 190 (2007), 1627–1636. [21] C. Ramachandran, H. M. Srivastava and A. Swaminathan, A unified class of k-uniformly convex functions defined by the Sălăjean derivative operator, Atti. Sem. Mat. Fis. Mod- ena Reggio Emilia 55 (2007), 47–59. [22] F. Rnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196. [23] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116. [24] H. Silverman, Univalent functions with varying arguments, Houston J. Math. 7 (1981), 283–287. [25] H. Silverman, A survey with open problems on univalent functions whose coefficients are negative, Rocky Mountain J. Math. 21 (1991), 1099–1125. [26] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math. 23 (1997), 169–174. REFERENCES 324 [27] J. Sokół, On some applications of the Dziok-Srivastava operator, Appl. Math. Comput. 201 (2008), 774–780. [28] H. M. Srivastava and M. K. Aouf, A certain fractional derivative operator and its appli- cations to a new class of analytic and multivalent functions with negative coefficients. I and II, J. Math. Anal. Appl. 171 (1992), 1–13; ibid. 192 (1995), 673–688. [29] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. [30] H. M. Srivastava and A. K. Mishra, Applications of fractional calculus to parabolic star- like and uniformly convex functions, Comput. Math. Appl. 39 (2000), 57–69. [31] H. M. Srivastava and S Owa, Certain classes of analytic functions with varying argu- ments, J. Math. Anal. Appl. 136 (1988), 217–228. [32] H. M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992. [33] H. M. Srivastava, T. N. Shanmugam, C. Ramachandran and S. Sivasubramanian, A new subclass of k-uniformly convex functions with negative coefficients, J. Inequal. Pure Appl. Math. 8 (2) (2007), Article 43, 1–14 (electronic). [34] Z.-G. Wang, Y.-P. Jiang and H. M. Srivastava, Some subclasses of multivalent analytic functions involving the Dziok-Srivastava operator, Integral Transform. Spec. Funct. 19 (2008), 129–146. [35] H. M. Srivastava, D.-G. Yang and N-E. Xu, Subordinations for multivalent analytic func- tions associated with the Dziok-Srivastava operator, Integral Transform. Spec. Funct. 20 (2009), 581–606. [36] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc. 12 (1961), 689–693.