Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied Mathematics INTEGRAL MEANS INEQUALITIES FOR FRACTIONAL DERIVATIVES OF SOME GENERAL SUBCLASSES OF ANALYTIC FUNCTIONS TADAYUKI SEKINE, KAZUYUKI TSURUMI, SHIGEYOSHI OWA AND H.M. SRIVASTAVA College of Pharmacy, Nihon University 7-1 Narashinodai 7-chome, Funabashi-shi Chiba , Japan Department of Mathematics Tokyo Denki University 2-2 Nisiki-cho, Kanda, Chiyoda-ku Tokyo , Japan. Department of Mathematics Kinki University, Higashi-Osaka Osaka , Japan. Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4 Canada. URL: c 2 School of Communications and Informatics,Victoria University of Technology ISSN (electronic): volume 3, issue 5, article 66, 22. Received 26 June, 22; accepted 4 July, 22. Communicated by: D.D. Bainov Abstract Home Page

2 Abstract Integral means inequalities are obtained for the fractional derivatives of order p + λ ( p n; λ < 1) of functions belonging to certain general subclasses of analytic functions. Relevant connections with various known integral means inequalities are also pointed out. 2 Mathematics Subject Classification: Primary 3C45; Secondary 26A33, 3C8. Key words: Integral means inequalities, Fractional derivatives, Analytic functions, Univalent functions, Extreme points, Subordination. The present investigation was initiated during the fourth-named author s visit to Saga National University in Japan in April 22. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP Introduction, Definitions, and Preliminaries The Main Integral Means Inequalities Remarks and Observations References Page 2 of 15

3 1. Introduction, Definitions, and Preliminaries Let A denote the class of functions f (z) normalized by f (z) = z + a k z k k=2 that are analytic in the open unit disk U := {z : z C and z < 1}. Also let A (n) denote the subclass of A consisting of all functions f (z) of the form: f (z) = z a k z k (a k ; n N := {1, 2, 3,...}). We denote by T (n) the subclass of A (n) of functions which are univalent in U, and by T α (n) and C α (n) the subclasses of T (n) consisting of functions which are, respectively, starlike of order α ( α < 1) and convex of order α ( α < 1) in U. The classes A(n), T (n), T α (n), and C α (n) were investigated by Chatterjea [1] (and Srivastava et al. [9]). In particular, the following subclasses: T := T (1), T (α) := T α (1), and C(α) := C α (1) were considered earlier by Silverman [7]. Page 3 of 15

4 Next, following the work of Sekine and Owa [4], we denote by A (n, ϑ) the subclass of A consisting of all functions f (z) of the form: (1.1) f (z) = z e i(k 1)ϑ a k z k (ϑ R; a k ; n N). Finally, the subclasses T (n, ϑ), Tα (n, ϑ), and C α (n, ϑ) of the class A (n, ϑ) are defined in the same way as the subclasses T (n), T α (n), and C α (n) of the class A (n). We begin by recalling the following useful characterizations of the function classes Tα (n, ϑ) and C α (n, ϑ) (see Sekine and Owa [4]). Lemma 1.1. A function f(z) A (n, ϑ) of the form (1.1) is in the class T α (n, ϑ) if and only if (1.2) (k α) a k 1 α (n N; α < 1). Lemma 1.2. A function f(z) A (n, ϑ) of the form (1.1) is in the class C α (n, ϑ) if and only if (1.3) k (k α) a k 1 α (n N; α < 1). Motivated by the equalities in (1.2) and (1.3) above, Sekine et al. [6] defined a general subclass A (n; B k, ϑ) of the class A (n, ϑ) consisting of functions f(z) Page 4 of 15

5 of the form (1.1), which satisfy the following inequality: B k a k 1 (B k > ; n N). Thus it is easy to verify each of the following classifications and relationships: and A (n; k, ϑ) = T (n, ϑ) =: T (n, ϑ) = T (n, ϑ), ( A n; k α ) 1 α, ϑ = Tα (n, ϑ) ( α < 1), ( ) k (k α) A n; 1 α, ϑ = C α (n, ϑ) ( α < 1). As a matter of fact, Sekine et al. [6] also obtained each of the following basic properties of the general classes A (n; B k, ϑ). Theorem 1.3. A (n; B k, ϑ) is the convex subfamily of the class A (n, ϑ). Theorem 1.4. Let (1.4) f 1 (z) = z and f k (z) = z ei(k 1)ϑ B k z k (k = n + 1, n + 2, n + 3,... ; n N). Then f A (n; B k, ϑ) if and only if f (z) can be expressed as follows: f (z) = λ 1 f 1 (z) + λ k f k (z), Page 5 of 15

6 where λ 1 + λ k = 1 (λ 1 ; λ k ; n N). Corollary 1.5. The extreme points of the class A (n; B k, ϑ) are the functions f 1 (z) and f k (z) (k n + 1; n N) given by (1.4). Applying the concepts of extreme points, fractional calculus, and subordination, Sekine et al. [6] obtained several integral means inequalities for higherorder fractional derivatives and fractional integrals of functions belonging to the general classes A(n; B k, ϑ). Subsequently, Sekine and Owa [5] discussed the weakening of the hypotheses for B k in those results by Sekine et al. [6]. In this paper, we investigate the integral means inequalities for the fractional derivatives of f(z) of a general order p + λ ( p n; λ < 1) of functions f(z) belonging to the general classes A(n; B k, ϑ). We shall make use of the following definitions of fractional derivatives (cf. Owa [3]; see also Srivastava and Owa [8]). Definition 1.1. The fractional derivative of order λ is defined, for a function f (z), by (1.5) Dz λ 1 d z f (ζ) f (z) := dζ ( λ < 1), Γ (1 λ) dz λ (z ζ) where the function f (z) is analytic in a simply-connected region of the complex z-plane containing the origin and the multiplicity of (z ζ) λ is removed by requiring log (z ζ) to be real when z ζ >. Page 6 of 15

7 Definition 1.2. Under the hypotheses of Definition 1.1, the fractional derivative of order n + λ is defined, for a function f (z), by Dz n+λ f (z) := dn dz n Dλ z f (z) ( λ < 1; n N := N {}). It readily follows from (1.5) in Definition 1.1 that (1.6) Dz λ z k Γ (k + 1) = Γ (k λ + 1) zk λ ( λ < 1). We shall also need the concept of subordination between analytic functions and a subordination theorem of Littlewood [2] in our investigation. Given two functions f (z) and g (z), which are analytic in U, the function f (z) is said to be subordinate to g (z) in U if there exists a function w (z), analytic in U with w () = and w (z) < 1 (z U), such that We denote this subordination by f (z) = g (w (z)) (z U). f (z) g (z). Theorem 1.6 (Littlewood [2]). If the functions f (z) and g (z) are analytic in U with g (z) f(z), then ( ) g re iθ µ ( dθ ) f re iθ µ dθ (µ > ; < r < 1). Page 7 of 15

8 2. The Main Integral Means Inequalities Theorem 2.1. Suppose that f(z) A (n; k p+1 B k, ϑ) and that (h + 1) q B h+1 Γ(h + 2 λ p) Γ(h + 1) Γ(n + 1 p) Γ(n + 2 λ p) B k (k n + 1) for some h n, λ < 1, and q p n. Also let the function f h+1 (z) be defined by (2.1) f h+1 (z) = z Then, for z = re iθ and < r < 1, (2.2) D p+λ z f (z) µ dθ e ihϑ (h + 1) q+1 B h+1 z h+1 ( fh+1 A ( n; k q+1 B k, ϑ )). D p+λ z f h+1 (z) µ dθ ( λ < 1; µ > ). Proof. By virtue of the fractional derivative formula (1.6) and Definition 1.2, we find from (1.1) that Dz p+λ f (z) = = z 1 λ p Γ (2 λ p) z 1 λ p Γ (2 λ p) ( ( 1 1 ) i(k 1)ϑ Γ(2 λ p)γ(k + 1) e a k z k 1 Γ(k + 1 λ p) ) e i(k 1)ϑ k! Γ(2 λ p) (k p 1)! Φ(k)a kz k 1, Page 8 of 15

9 where (2.3) Φ(k) := Γ (k p) Γ (k + 1 λ p) ( λ < 1; k n + 1; n N). Since Φ(k) is a decreasing function of k, we have Γ (n + 1 p) < Φ (k) Φ (n + 1) = Γ(n + 2 λ p) ( λ < 1; k n + 1; n N). Similarly, from (2.1), (1.6), and Definition 1.2, we obtain, for λ < 1, Dz p+λ f h+1 (z) = z 1 λ p Γ (2 λ p) ( 1 For z = re iθ and < r < 1, we must show that 1 e ihϑ Γ(2 λ p)γ(h + 2) (h + 1) q+1 B h+1 Γ(h + 2 λ p) e i(k 1)ϑ k! Γ(2 λ p) (k p 1)! Φ(k)a kz k 1 1 e ihϑ Γ(2 λ p)γ(h + 2) µ z h dθ, (h + 1) q+1 B h+1 Γ(h + 2 λ p) µ dθ z h ). ( λ < 1; µ > ). Page 9 of 15

10 Thus, by applying Theorem 1.6, it would suffice to show that (2.4) 1 Indeed, by setting 1 we find that e i(k 1)ϑ k! Γ(2 λ p) (k p 1)! Φ(k)a kz k 1 1 e ihϑ Γ(2 λ p)γ(h + 2) z h. (h + 1) q+1 B h+1 Γ(h + 2 λ p) e i(k 1)ϑ k! Γ(2 λ p) (k p 1)! Φ(k)a kz k 1 = 1 e ihϑ Γ(2 λ p)γ(h + 2) {w(z)} h, (h + 1) q+1 B h+1 Γ(h + 2 λ p) {w(z)} h = (h + 1)q+1 B h+1 Γ(h + 2 λ p) e ihϑ Γ(h + 2) e i(k 1)ϑ k! (k p 1)! Φ(k)a kz k 1, which readily yields w() =. Therefore, we have w(z) h (h + 1)q+1 B h+1 Γ(h + 2 λ p) Γ(h + 2) k! (k p 1)! Φ(k)a k z k 1 Page 1 of 15

11 (2.5) z n (h + 1)q+1 B h+1 Γ(h + 2 λ p) Γ(h + 2) k! Φ(n + 1) (k p 1)! a k = z n (h + 1)q+1 B h+1 Γ(h + 2 λ p) Γ(h + 2) Γ(n + 1 p) Γ(n + 2 λ p) = z n (h + 1)q B h+1 Γ(h + 2 λ p) Γ(h + 1) Γ(n + 1 p) Γ(n + 2 λ p) z n z n k! (k p 1)! B ka k k p+1 B k a k z n < 1 (n N), k! (k p 1)! a k k! (k p 1)! a k by means of the hypothesis of Theorem 2.1. In light of the last inequality in (2.5) above, we have the subordination (2.4), which evidently proves Theorem 2.1. Page 11 of 15

12 3. Remarks and Observations First of all, in its special case when p = q =, Theorem 2.1 readily yields Corollary 3.1 (cf. Sekine and Owa [5], Theorem 6). Suppose that f(z) A (n; kb k, ϑ) and that B h+1 Γ(h + 2 λ) Γ(h + 1) Γ(n + 1) Γ(n + 2 λ) B k (k n + 1; n N) for some h n and λ < 1. Also let the function f h+1 (z) be defined by (3.1) f h+1 (z) = z Then, for z = re iθ and < r < 1, (3.2) D λ z f (z) µ dθ e ihϑ (h + 1)B h+1 z h+1 (f h+1 A (n; kb k, ϑ)). D λ z f h+1 (z) µ dθ ( λ < 1; µ > ). A further consequence of Corollary 3.1 when h = n would lead us immediately to Corollary 3.2 below. Corollary 3.2. Suppose that f(z) A (n; kb k, ϑ) and that (3.3) B n+1 B k (k n + 1; n N). Also let the function f n+1 (z) be defined by f n+1 (z) = z e inϑ (n + 1)B n+1 z n+1 (f h+1 A (n; kb k, ϑ)). Page 12 of 15

13 Then, for z = re iθ and < r < 1, D λ z f (z) µ dθ D λ z f n+1 (z) µ dθ ( λ < 1; µ > ). The hypothesis (3.3) in Corollary 3.2 is weaker than the corresponding hypothesis in an earlier result of Sekine et al. [6, p. 953, Theorem 6]. Next, for p = 1 and q =, Theorem 2.1 reduces to an integral means inequality of Sekine and Owa [5, Theorem 7] which, for h = n, yields another result of Sekine et al. [6, p. 953, Theorem 7] under weaker hypothesis as mentioned above. Finally, by setting p = q = 1 in Theorem 2.1, we obtain a slightly improved version of another integral means inequalities of Sekine and Owa [5, Theorem 8] with respect to the parameter λ (see also Sekine et al. [6, p. 955, Theorem 8] for the case when h = n, just as we remarked above). Page 13 of 15

14 References [1] S.K. CHATTERJEA, On starlike functions, J. Pure Math., 1 (1981), [2] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc. (2), 23 (1925), [3] S. OWA, On the distortion theorems. I, Kyungpook Math. J., 18 (1978), [4] T. SEKINE AND S. OWA, Coefficient inequalities for certain univalent functions, Math. Inequal. Appl., 2 (1999), [5] T. SEKINE AND S. OWA, On integral means inequalities for generalized subclasses of analytic functions, in Proceedings of the Third ISAAC Congress, Berlin, August 21. [6] T. SEKINE, K. TSURUMI AND H.M. SRIVASTAVA, Integral means for generalized subclasses of analytic functions, Sci. Math. Japon., 54 (21), [7] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), [8] H.M. SRIVASTAVA AND S. OWA (Eds.), Univalent, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, Page 14 of 15

15 [9] H.M. SRIVASTAVA, S. OWA AND S.K. CHATTERJEA, A note on certain classes of starlike functions, Rend. Sem. Mat. Univ. Padova, 75 (1987), Page 15 of 15