Research Article Coefficient Inequalities for a Subclass of p-valent Analytic Functions

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1 e Scientific World Journal, Article ID , 5 pages Research Article Coefficient Inequalities for a Subclass of p-valent Analytic Functions Muhammad Arif, 1 Janusz SokóB, and Muhammad Ayaz 1 1 Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 300, Pakistan Department of Mathematics, RzeszówUniversityofTechnology,Al.PowstancowWarszawy1,35-959Rzeszów, Poland Correspondence should be addressed to Muhammad Arif; marifmaths@yahoo.com Received 31 August 013; Accepted 19 November 013; Published 4 February 014 Academic Editors: Y.-M. Chu, A. I. Delis, B. Dragovich, A. Fošner, and B. Meng Copyright 014 Muhammad Arif 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. The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass of p-valent analytic functions. Many known results appear as special consequences of our work. 1. Introduction Let A(p) denote the class of functions f(z) of the form f (z) =z p + a n z n, (p N), (1) which are analytic and multivalent in the open unit disk U = {z : z < 1}. AlsoletS p and K p denote the well-known classes of p-valent starlike functions and p-valent convex functions, respectively. For f(z) A(p) given by (1)andg(z) A(p) given by Motivated by Ruscheweyh operator [1], Goel and Sohi [] introduced a differential operator D δ+p 1 for p-valent analytic functions given by D δ+p 1 z f (z) = (1 z) f(z) δ+p =zp + with δ> p, φ n (δ) a n z n, (4) φ n (δ) = (δ + p) n p, (5) (n p)! and (x) n is a Pochhammer symbol given by g (z) =z p + b n z n, (p N), () 1, n = 0, (x) n ={ x (x+1)(x+) (x+n 1), n N. It is obvious that when δ is any integer greater than p, (6) the Hadamard product (or convolution) of f(z) and g(z) is given by (f g) (z) =z p + a n b n z n, (z U). (3) D δ+p 1 f (z) = zp (z δ 1 f (z)) (δ+p 1) (δ+p 1)!. (7) The following identity can be easily established: (δ+p) D δ+p f (z) =δd δ+p 1 f (z) +z(d δ+p 1 f (z)). (8)

2 The Scientific World Journal Using the generalized Ruscheweyh operator, we define a subclass of VD λ p (δ,b,α,β)of p-valent analytic functions as follows. Definition 1. An analytic p-valent function f(z) of the form (1)belongstotheclassVD λ p (δ,b,α,β),ifandonlyif Re {e iλ (1 b + D δ+p f (z) b D δ+p 1 f (z) )} >α b ( Dδ+p f (z) D δ+p 1 f (z) 1) +βcos λ, (9) Proof. Let f(z) VD λ p (δ,b,α,β).thenweobtain Re e iλ {1 b + D δ+p f (z) b D δ+p 1 f (z) } >α b ( Dδ+p f (z) D δ+p 1 f (z) 1) +βcos λ >αre e iλ (1 b + D δ+p f (z) b D δ+p 1 f (z) ) αre e iλ +βcos λ, and this implies (10) where α 0, b C \{0}, δ> p, λ is real with λ < (π/), and 0 β<1. By giving specific values to α, β, λ, p, b, and δ in VD λ p (δ,b,α,β), we obtain many important subclasses studied by various authors in earlier papers; see for details [3 6]; we list some of them as follows: (i) VD λ 1 (0,,0,0) S λ and VDλ 1 (1,1,0,0) K λ, studied by Spacek [7]and Robertson[8], respectively; for the advancement work see [9 11]; (ii) VD 0 1 (0,,α,β) SD(α, β) and VD0 1 (1,1,α,β) KD(α, β), studied by both Owa et al. and Shams et al. [1, 13]; (iii) VD λ 1 (0,,1,0) USP(λ) and VDλ 1 (1,1,1,0) UCSP(λ), introduced by Ravichandran et al. [14]; (iv) VD 0 1 (δ,b,α,β) VD(δ,b,α,β), considered by Latha [15]; (v) VD 0 1 (0,,0,β) S (β) and VD 0 1 (1,1,0,β) K(β), the well-known classes of starlike and convex functions of order β. From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions. We will assume throughout our discussion, unless otherwise stated, that α 0, 0 β<1, δ> 1, λ is real with λ < (π/),andb C \{0}. Re [e iλ (1 b + D δ+p f (z) )] > ζ cos λ. (11) b D δ+p 1 f (z) Also if 0 α β,thenwecaneasilyobtain and this completes the proof. 0 ζ<1, (1) Theorem 3. If f(z) VD λ p (δ,b,α,β),then and a n+p 1 (δ + p) b η (n 1) 1 α φ n+p 1 (δ) n where φ n+p 1 (δ) is given by (5) and a b η 1 α, (13) (1 + (δ + p) b η ), n 3, j 1 α (14) η=(1 β)cos λ+i(1 α) sin λ. (15) Proof. Let f(z) VD λ p (δ,b,α,β).thenbytheorem, we have Let us define p(z) by Re [e iλ (1 b + D δ+p f (z) b D δ+p 1 f (z) )] >( β α ) cos λ, (z E). 1 α (16). Main Results Theorem. Let f(z) VD λ p (δ,b,α,β)with 0 α β.then f(z) VD λ p (δ,b,0,ζ),whereζ = (β α)/(1 α). e iλ (1 b + D δ+p f (z) b D δ+p 1 f (z) ) =[( 1 β )p(z) +(β α)] cos λ+isin λ. 1 α 1 α (17)

3 The Scientific World Journal 3 Then p(z) is analytic in E with p(0) = 1 and Re p(z) > 0, z E.Let Then (17)becomes p (z) =1+ p n z n, z E. (18) 1 b + D δ+p f (z) b D δ+p 1 f (z) (1 β) cos λ+i(1 α) sin λ =1+ e iλ (1 α) That is, e iλ (1 α) [D δ+p f (z) D δ+p 1 f (z)] =bηd δ+p 1 f (z) p n z n, p n z n. where η is given by (15). Using (8)in(0), we obtain (19) (0) which shows that (13)istrue.Forn=3, a p+ b η (δ + p) 1 α () φ p+ (δ) {1+φ p+1 (δ) a p+1 }, (6) and using (13), we have a p+ b η (δ + p) (1 α)() φ p+ (δ) {1 + b η (δ + p) }. (7) (1 α) Therefore, (14) holdsforn=3. Assume that (14)is true for n=k;that is, a k+p 1 b η k (δ + p) (1 + b η (δ + p) ). 1 α (k 1) φ k+p 1 (δ) 1 α j (8) Consider a k+p b η (δ + p) 1 α (k) φ k+p (δ) e iλ (1 α) [z(d δ+p 1 f(z)) pd δ+p 1 f (z)] or, equivalently, e iλ (1 α)[ [ =b(δ+p)ηd δ+p 1 f (z) p n z n, =b(δ+p)η[ z p + [ (k p) φ k (δ) a k z k ] ] φ k (δ) a k z k ] ] ( p n z n ). (1) () (δ + p) { { (1 + b η 1 α { (1+ b η 1 α (δ + p) ) + + b η (δ + p) 1 α (k 1) k (1 + b η (δ + p) 1 α j k 1 )+ b η (δ + p) 1 α () ) } } } b = η (δ + p) (1 + b η (δ + p) ). 1 α (k) φ k+p (δ) 1 α j (9) Comparing the coefficients of z n+p 1 on both sides, e iλ (1 α)(n 1) φ n+p 1 (δ) a n+p 1 =b(δ+p)η{p 1 a n+p φ n+p (δ) + +p n 1 }. (3) Taking absolute on both sides and then applying the coefficient estimates p n for Caratheodory functions [3], we have a n+p 1 b (δ + p) η (n 1) 1 α φ n+p 1 (δ) {1+φ p+1 (δ) a p+1 + +φ n+p (δ) a n+p }. (4) We apply mathematical induction on (4). So for n=, a p+1 b η 1 α, (5) Therefore, the result is true for n=k+1, and hence by using mathematical induction, (14) holds true for all n 3. If we put λ=0, p=1, b=,andδ=0in Theorem 3,we obtaintheresultprovedin[1]. Corollary 4. If f(z) SD(α, β),then a (1 β), 1 α n a n (1 β) (1 + (1 β) ), (n 3). (n 1) 1 α j 1 α (30) If one takes α = 0 in Corollary 4, one obtains the following inequality: n a n 1 (j β), (n ), (31) (n 1)! j= which was proved by Robertson [16].

4 4 The Scientific World Journal By setting λ=0, p=1, b=1,andδ=1in Theorem 3, one obtains the result proved in [1]. Corollary 5. If f(z) KD(α, β),then a n (1 β) 1 α, a n (1 β) (1 + (1 β) ), (n 3). n (n 1) 1 α j 1 α (3) Using (8) and then simplifications gives Q (z) 1 = ({bδ(eiλ 1)+bp(e iλ 1) pe iλ } D δ+p 1 f (z) +e iλ z(d δ+p 1 f (z)) ) (b(δ+p)d δ+p 1 f (z)) 1 ( (δ + p) b sin λ + k (δ) a k z φ k (37) Letting α=0in Corollary 5, one gets the following inequality proved by Robertson [16]: n a n 1 (j β), (n ). (33) n! j= Theorem 6. If f(z) A(p) and satisfies [(δ+p) b sin λ +(k p)]) ((δ+p) b (1 Now consider α Q (z) 1 Re Q (z) 1 +βcos λ (α+1) Q (z) 1 +βcos λ φ k (δ) a k )) 1. (α+1)(δ + p) b sin λ +β(δ+p) b cos λ + φ k (δ) a k [(α+1) { (δ + p) b sin λ + (k p)} β (δ + p) b cos λ+(δ+p) b ] (34) (((α+1) { { sin {(δ+p) b λ + k (δ) a k z φ k ((δ + p) b (1 [(δ+p) b sin λ +(k p)]} }) } φ k (δ) a k )) 1) <(δ+p) b, where φ k (δ) is given by (5),thenf(z) VD λ p (δ,b,α,β). Proof. Suppose (34)holds.Alsoletussuppose Q (z) =e iλ (1 b + D δ+p f (z) ). (35) b D δ+p 1 f (z) +βcos λ. The last expression is bounded by 1 if (α+1)(δ + p) b sin λ +β(δ+p) b cos λ + φ k (δ) a k [(α+1) { (δ + p) b sin λ +(k p)} (38) Then Q (z) 1 (be iλ e iλ b)d δ+p 1 f (z) +e iλ D δ+p f (z) =. bd δ+p 1 f (z) (36) β(δ+p) b cos λ+(δ+p) b ]<(δ+p) b, (39) and this completes the proof. For λ=0, δ= p+1, b=,andα=0in Theorem 6,we obtain the following.

5 The Scientific World Journal 5 Corollary 7. If f(z) A(p) and satisfies (k+1 p β) a k <(1 β), (40) then f(z) S p (β), theclassofp-valent starlike functions of order β. For λ=0, δ= p+, b=1,andα=0in Theorem 6,one has the following. Corollary 8. If f(z) A(p) and satisfies (k p+1)(k+1 p β) a k <(1 β), (41) then f(z) K p (β), theclassofp-valent convex functions of order β. Further for p = 1 in both the last two corollaries, one obtains the results for the classes S (β) and K(β) which was proved by Merkes et al. [17] and Silverman[18], respectively. Conflict of Interests The authors declare that they have no conflict of interests. Please consider this paper for further process. References [11] K. I. Noor, M. Arif, and A. Muhammad, Mapping properties of some classes of analytic functions under an integral operator, JournalofMathematicalInequalities,vol.4,no.4,pp , 010. [1] S. Owa, Y. Polatoǧlu, and E. Yavuz, Coefficient inequalities for classes of uniformly starlike and convex functions, Inequalities in Pure and Applied Mathematics,vol.7,no.5,article 160, 006. [13] S. Shams, S. R. Kulkarni, and J. M. Jahangiri, Classes of uniformly starlike and convex functions, International Journal of Mathematics and Mathematical Sciences,vol.004,no.55,pp , 004. [14] V. Ravichandran, C. Selvaraj, and R. Rajagopal, On uniformly convex spiral functions and uniformly spirallike function, Soochow Mathematics, vol.9,no.4,pp , 003. [15] S. Latha, Coefficient inequalities for certain classes of ruscheweyh type analytic functions, Inequalities in Pure and Applied Mathematics,vol.9,no.,article5,008. [16] M. S. Robertson, On the theory of univalent functions, Annals of Mathematics,vol.37,pp ,1936. [17] E.P.Merkes,M.S.Robertson,andW.T.Scott, Onproductsof starlike functions, Proceedings of the American Mathematical Society,vol.13,pp ,196. [18] H. Silverman, Univalent functions with negative coeffcients, Proceedings of the American Mathematical Society, vol.51,pp , [1] S. Ruscheweyh, A new criteria for univalent function, Proceedings of the American Mathematical Society,vol.49,no.1,pp , [] R. M. Goel and N. Sohi, A new criteria for p-valent functions, Proceedings of the American Mathematical Society, vol.78,pp , [3] E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, Classes of k- uniformly convex and starlike functions, Tamkang Mathematics,vol.35,no.3,pp.1 7,004. [4] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, Computational and Applied Mathematics, vol. 105, no. 1-, pp , [5] S. Kanas and A. Wisniowska, Conic domains and starlike functions, RevueRoumainedeMathématique Pures et Appliquées, vol. 45, pp , 000. [6] J. Nishiwaki and S. Owa, Certain classes of analytic functions concerned with uniformly starlike and convex functions, Applied Mathematics and Computation,vol.187,no.1,pp , 007. [7] L. Spacek, Prispĕvek k teorii funkei prostych, Časopis pro Pěstování Matematiky a Fysik,vol.6,pp.1 19,1933. [8] M.S.Robertson, Univalentfunctionsf(z) for wich zf (z) is spiral-like, Michigan Mathematical Journal, vol.16,pp , [9] M. Arif, On certain suffciency criteria for p-valent meromorphic spiralike functions, Abstract and Applied Analysis, vol. 013,ArticleID837913,9pages,013. [10] M. Arif, K. I. Noor, M. Raza, and W. Haq, Some properties of a generalized class of analytic functions related with Janowski functions, Abstract and Applied Analysis, vol.01,articleid 79843, 11 pages, 01.

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