Research Article Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative

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1 Hindawi Function Spaces Volume 217 Article ID pages Research Article Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative Xiaofei Li 1 Deng Ding 1 Liping Xu 2 Chuan Qin 3 and Songbo Hu 4 1 Department of Mathematics University of Macau Taipa Macau 2 School of Information and Mathematics Yangtze University Jingzhou 434 China 3 Engineering and Technology College Yangtze University Xueyuan Rd. Jingzhou 434 China 4 School of Health Sciences Wuhan University Wuhan 4372 China Correspondence should be addressed to Liping Xu; xlp211@126.com Received 3 September 216; Accepted 21 November 216; Published 17 January 217 Academic Editor: Enrique Llorens-Fuster Copyright 217 Xiaofei Li et al. 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. In this paper we define and study some subclasses of multivalent analytic functions of higher order in the unit disc. These classes generalize some classes previously studied. We obtain coefficient inequalities distortion theorems extreme points and integral mean inequalities. We derive some results as special cases. 1. Introduction Let p N = {12...} and denote A p as the class of multivalent functions of the form f (z) =z p + a n z n (1) which are analytic in the open unit disk: U = {z C : z <1}. (2) For two parameters α [ p p] and β function f(z) A p is said to be in class UST(p α β) of p-valent β-uniformly star-like functions of order α in Uifandonlyif R ( zf (z) f (z) zf (z) f (z) α) β p (3) where R( ) denotes taking the real part of argument. On the other hand function f(z) A p is said to be in class UCV(p α β) of p-valent β-uniformly convex functions of order α in Uifandonlyif R (1 + zf (z) f (z) α) β 1+ zf (z) f (z) p. (4) We note from (3) and (4) that f (z) UCV (p α β) zf (z) p UST (pαβ). The classes UST(p α β) and UCV(p α β) were introduced recently by Khairnar and More [1]. Various subclasses of analytic and univalent or multivalent functions were studied in many papers (see e.g. [2 4]). Recently Nishiwaki and Owa in [5] introduced two classes MD(α β) consisting of all functions f(z) A 1 whichsatisfy R ( zf (z) f (z) (5) zf (z) α)<β 1 (6) f (z) and ND(α β) consisting of all functions f(z) A 1 which satisfy R (1 + zf (z) f (z) α)<β zf (z) f (7) (z)

2 2 Function Spaces where α 1and β. We notice from definitions of these classes that f (z) ND (α β) (8) zf (z) MD (α β). For each f(z) A p it is easily seen upon differentiating both sides of (1) q times with respect to z that f (q) (z) =δ(pq)z p q + δ(nq)a n z n q (9) where p > q N = N {}andδ(p q) denotes qpermutationsof p objects; that is p! δ (p q) = (p q)!. (1) Let q m N and p Nsuch that p>q+m and assume that δ (p qm) α<δ(p qm) β. (11) Srivastava et al. in [6] introduced a subclass of the p-valent function class US m (pq;αβ)consisting of functions f(z) of form (1) which satisfy the following analytic criterion: for all z U R ( zm f (q+m) (z) α) f (q) (z) (12) z m f (q+m) (z) β δ(p qm). f (q) (z) Recently many papers have discussed such aspects of analytic univalent or multivalent functions. For example Aouf et al. in [7 8] discussed a subclass of p-valent analytic function with negative coefficient by using higher-order derivatives andinvestigatedmanypropertiesofdistortiontheorems closure theorems modified Hadamard products and radii of close-to-convexity starlikeness and convexity. K. I. Noor andm.a.noorin[9]definedsomesubclassesofanalytic functions related to k-uniformly close-to-convex functions of higher order and studied the following: rate of growth of coefficients inclusion relations radius problems and necessary conditions for univalency. Seoudy in [1] defined a class of p-valent functions defined by certain linear operator and obtained subordination and inclusion properties. Following Srivastava et al. [6] and Nishiwaki and Owa [5] we define a new subclass MD m (pq;αβ)of multivalent functions involving higher-order derivative. Definition 1. Let q m N and p N such that p>q+m and α and β two real numbers satisfy that α δ(p qm) and β.functionf(z) MD m (p q; α β) ifandonlyif f(z) A p and satisfies the following inequality: R ( zm f (q+m) (z) f (q) (z) α) z m f (q+m) (z) <β δ(p qm). f (q) (z) (13) From the above definition it is clear that MD 1 (1 ; α β) = MD(α β). In this paper we obtain several properties including the coefficient inequalities distortion theorems extreme points and integral means inequalities for this subclass MD m (pq;αβ)of multivalent functions involving higher-order derivative. 2. Coefficient Inequalities We derive sufficient conditions for f(z) which are given by using coefficient inequalities. Theorem 2. Let f(z) be a function of form (1). If the coefficients of f(z) satisfy where Φ(npqmαβ) a n (14) δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ)=δ(nq) [ δ(n qm)+δ(p qm) α + δ(n qm) δ(p qm) α 2β δ(n qm) δ(p qm) ] then f(z) is in class MD m (pq;αβ). Proof. Suppose that inequality (14) holds and denote F (z) = zm f (q+m) (z) f (q) (z) α z m f (q+m) (z) β δ(p qm). f (q) (z) From the definition we can verify that if (15) (16) F (z) +δ(p qm) <1 (17) F (z) δ(p qm) then f(z) MD m (pq;αβ). In fact denoting Ψ ± (z; f m p q α β) = z m f (q+m) (z) αf (q) (z) ±δ(p qm)f (q) (z) βe iθ zm f (q+m) (z) δ(p qm)f (q) (z) Σ ± (z; m p q α) = δ(nq) (δ(n qm)±δ(p qm) α)a n z n p Ξ ± (mpqα)= δ(nq) δ(n qm)±δ(p qm) α a n (18)

3 Function Spaces 3 we have F (z) +δ(p qm) Ψ + (z;fmpqαβ) = F (z) δ(p qm) Ψ (z;fmpqαβ) δ(pq)(2δ(p qm) α)+σ + (z; m p q α) βe iθ = Σ (z; m p q ) [αδ (p q) Σ (z;mpqα)+βe iθ Σ (z; m p q ) ] δ(pq) 2δ (p q m) α +Ξ+ (mpqα) βξ (mpq). αδ (p q) Ξ (mpqα)+βξ (mpq) (19) Here we use technology f(z) = e iθ f(z). If (14) satisfies we drive that the last expression above is bounded by 1 which implies f(z) MD m (p q; α β). Thustheproofof Theorem 2 is completed. Example 3. Function f(z) given by f (z) =z p + δ(pq)(α 2δ (p q m) α )(γ+p+1)ξ (2) n n z (n + γ) (n + γ + 1) Φ (n p q m α β) belongs to class MD m (p q; α β) for γ> (p+1)and ξ n C with ξ n =1. As a special case of Theorem 2 as in [2] we can obtain the following corollary. Corollary 4. Function f(z) A 1 is in class MD(α β)if n=2 [ n+1 α + n 1 α 2β(n 1)] a n α 2 α. (21) In view of Theorem 2 we introduce subclass MD m (p q; α β) which consists of functions of the form f (z) =z p + a n z n (22) whose Taylor-Maclaurin coefficients a n are nonnegative and satisfy inequality (14). By the coefficient inequalities for classes MD m (pq;αβ) we have the following theorem. Theorem 5. If β 1 β 2 then MD m (p q; α β 1) MD m (p q; α β 2). (23) Since MD 1 (1;αβ) = MD(α β)wegetthefollowing corollary which is a theorem in [2]. Corollary 6. If β 1 β 2 then 3. Distortion Theorems MD (α β 1 ) MD (α β 2 ). (24) Lemma 7. If f(z) MD m (p q; α β) then there exists N such that where a n A p (25) A p = δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ)a n (26) and Φ(n p q m α β) is given in (15). Proof. From the definition of Φ(n p q m α β) there exists N such that function Φ(n p q m α β) is increasing with respect to n when n>. According to Theorem 2 we have Φ(npqmαβ)a n δ(pq)(α 2δ (p q m) α ) From Φ (n p q m α β) a n. a n Φ(npqmαβ)a n (27) (28)

4 4 Function Spaces we have a n δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ)a n. (29) Using the same argument we obtain the following inequality. Lemma 8. If f(z) MD m (pq;αβ) then there exists N such that na n B p (3) This implies that inequality (25) holds. where B p = ( +1)[δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ)a n] (31) and Φ(n p q m α β) is defined by (15). Theorem 9. Let f(z) be a function in class MD m (p q; α β). Then for z = r < 1 f (z) rp + f (z) rp a n r n +A p r +1 a n r n A p r +1 (32) where A p and B p are given in Lemmas 7 and 8 respectively. Proof. Let f(z) be a function of form (22). For z =r<1 using Lemma 7 we have f (z) z p + z p + r p + f (z) z p a n z n + a n z n + z +1 a n r n +A p r +1 a n z n a n z n a n a n z n (33) Theorem 1. Let f(z) be a function in class MD m (pq;αβ). Then for z = r < 1 f (z) prp 1 + f (z) prp 1 na n r n 1 +B p r na n r n 1 B p r where A n and B n aregiveninlemmas7and8respectively. 4. Extreme Points (34) Theorem 11. Let f p (z) = z p and for each n = p + 1 p define f n (z) =z p + δ(pq)(α 2δ (p q m) α ) z n (35) Φ(npqmαβ) where Φ(n p q m α β) is defined by (15). Then f(z) (pq;αβ)if and only if it can be expressed in the form MD m f (z) = n=p λ n f n (z) (36) where λ n for all n=pp+1...and n=p λ n =1. z p r p a n z n z +1 a n r n A p r +1. a n Using the same argument we can prove the following result. Proof. Suppose that f (z) = =z p + n=p λ n f n (z) =λ p f p (z) + λ n f n (z) δ(pq)(α λ 2δ (p q m) α ) n z n. Φ(npqmαβ) (37)

5 Function Spaces 5 Then Φ(npqmαβ)λ n δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ) = λ n δ(pq)(α 2δ (p q m) α ) =δ(pq)(α 2δ (p q m) α )(1 λ p) δ(pq)(α 2δ (p q m) α ). (38) Thus it follows from Theorem 2 that f(z) MD m (pq;αβ). Conversely suppose that f(z) MD m (pq;αβ).since a n δ(pq)(α 2δ (p q m) α ) Φ(npqmαβ) we denote λ n = Φ(npqmαβ) δ(pq)(α 2δ (p q m) α )a n p+2... p+2... (39) (4) And λ p =1 λ n. By a simple calculation we get f(z) MD m (pq;αβ). Corollary 12. The extreme points of MD m (pq;αβ) are functions f p (z) = z p and f n (z) =z p + δ(pq)(α 2δ (p q m) α ) z n (41) Φ(npqmαβ) for each p Integral Means Inequalities Assume that two functions f(z) and g(z) are analytic in U. We say that f(z) is subordinate to g(z) written as f(z) g(z)ifthereexistsananalyticfunctionw(z) in U with w() = 1 and w(z) < 1 such that f(z) = g(w(z)). Lemma 13 (see [11]). If f(z) and g(z) are analytic in U with f(z) g(z)thenforμ>and z=re iθ (<r<1) f (z) μ dθ g (z) μ dθ. (42) Theorem 14. Let f(z) MD m (pq;αβ)and f n(z) be given by (35). Suppose that Φ(npqmαβ) a n δ(pq)(α 2δ (p q m) α ). (43) If there exists function w(z) z U that satisfied the condition Φ(npqmαβ) w (z) =( δ(pq)(α 2δ (p q m) α ) (44) a n z n p ) 1/(n p) then for z=re iθ < r < 1wehave f (z) μ dθ f n (z) μ dθ. (45) Proof. In order to obtain the result it is necessary to prove the following inequality: 1+ a n z n p μ dθ 1 + δ(pq)(α 2δ (p q m) α ) μ z n p dθ. Φ(npqmαβ) From Lemma 13 it is sufficient to verify the subordination: 1+ a n z n p 1+ δ(pq)(α 2δ (p q m) α ) z n p. Φ(npqmαβ) Thus there exists an analytic function w(z) in C such that 1+ a n z n p =1+ δ(pq)(α 2δ (p q m) α ) w (z) n p. Φ(npqmαβ) We find that Φ(npqmαβ) w (z) =( δ(pq)(α 2δ (p q m) α ) a n z n p ) 1/(n p) which readily yields w() = and w (z)1/(n p) Φ(npqmαβ) = δ(pq)(α 2δ (p q m) α ) n z a n p Φ(npqmαβ) δ(pq)(α 2δ (p q m) α ) a n zn p Φ(npqmαβ) δ(pq)(α 2δ (p q m) α ) z z <1. a n (46) (47) (48) (49) (5)

6 6 Function Spaces This means that the hypotheses of w(z) aresatisfiedandthe theorem is proved. Competing Interests The authors declare that there are no competing interests regarding the publication of this paper. Acknowledgments ThisworkispartiallysupportedUniversityofMacauwiththe Research Grant (MYRG68(Y1-L2)-FST13-DD) the research fund from Engineering and Technology College Yangtze University (no. 15J82) and research fund from Wuhan University (no KF29). References [1] S. M. Khairnar and M. More On a subclass of multivalent β-uniformly starlike and convex functions defined by a linear operator IAENG International Applied Mathematics vol.39no.3pp [2] S. Shams S. R. Kulkarni and J. M. Jahangiri Classes of uniformly starlike and convex functions International Journal of Mathematics and Mathematical Sciencesvol.24no.55pp [3] O. Altıntaş H. Irmak and H. M. Srivastava Neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator Computers & Mathematics with Applicationsvol.55no.3pp [4] N. Breaz and R. M. El-Ashwah Quasi-Hadamard product of some uniformly analytic and p -valent functions with negative coefficients Carpathian Mathematics vol. 3 no. 1 pp [5] J. Nishiwaki and S. Owa Certain classes of analytic functions concerned with uniformly starlike and convex functions Applied Mathematics and Computationvol.187no.1pp [6] H. M. Srivastava R. M. El-Ashwah and N. Breaz A certain subclass of multivalent functions involving higher-order derivatives Filomat vol. 3 no. 1 pp [7] M. K. Aouf T. M. Seoudy and G. M. El-Hawsh Certain subclass of higher order derivatives of p-valent functions International Open Problems in Complex Analysisvol. 7 no. 1 pp [8] M. K. Aouf A. O. Mostafa and W. K. Elyamany Certain subclass of multivalent functions with higher order derivatives and negative coefficients International Open Problems in Complex Analysisvol.8pp [9] K. I. Noor and M. A. Noor Higher order uniformly close-toconvex functions Discrete and Continuous Dynamical Systems Series Svol.8no.6pp [1] T. M. Seoudy Some applications of subordination for higherorder derivatives of p -valent functions defined by certain linear operator New Zealand Mathematics vol. 44 pp [11] J. E. Littlewood On inequalities in the theory of functions Proceedings of the London Mathematical Society vol.23no.1 pp

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