ON A SUBCLASS OF ANALYTIC CLOSE TO CONVEX FUNCTIONS IN q ANALOGUE ASSOCIATED WITH JANOWSKI FUNCTIONS. 1. Introduction and definitions

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1 Journal of Classical Analysis Volume 13, Number , doi: /jca ON A SUBCLASS OF ANALYTIC CLOSE TO CONVEX FUNCTIONS IN q ANALOGUE ASSOCIATED WITH JANOWSKI FUNCTIONS BAKHTIAR AHMAD, MUHAMMAD FAROOQ AND RAEES KHAN Abstract. In this article we define a new subclass of analytic multivalent close-to-convex functions in q-calculus associated with Janowski functions. We investigate some geometric properties such as sufficiency criteria, distortion problem, growth theorem, radii of starlikeness and convexity and coefficient estimates for this class. 1. Introduction and definitions The q-calculus, which is calculus without limits, has attracted the mathematicians due to its numerous physical and mathematical applications. The generalization of derivative and integral in q-calculus which are known as q-analogue of derivative and integral were introduced and studied by Jackson [11, 12]. Aral and Gupta [4, 5, 6]used this concept and defined the q-baskakov Durrmeyer operator using q-beta function. Similarly, the authors in [3, 7] gave the generalizationof some complex operators called q-picard and q-gauss-weierstrass singular integral operators. Srivastava and Bansal [19, pp. 62] used this concept in Geometric function theory and introduced the q- generalization of starlike functions for the first time, see also [18, pp. 347 et seq.]. In 2014, the q-analogue of Ruscheweyh operators were studied by Kanas and Răducanu [14], and investigated their properties. Later Mohammed and Darus [2] and Mahmood and Sokół [15] studied this differential operator. In this article we introduce the q-analogue of a subclass of close-to-convex multivalent functions in association with Janowski functions and study its geometric properties like sufficiency criteria, coefficient bounds, radii problems and distortion theorem. Let A p denote the class of all analytic multivalent functions f that are analytic in the open unit disc D = {z C : z < 1} and satisfying the normalization f z=z p + k=1 where p is a positive integer. The q-derivative of a function f is defined by a k+p z k+p, z D, 1 q f z= f qz f z, z 0, 2 zq 1 Mathematics subject classification 2010: 30C45, 30C50. Keywords and phrases: Multivalent analytic functions, starlike functions, close-to-convex functions, Janowski functions. c D l,zagreb Paper JCA

2 84 B. AHMAD,M.FAROOQ AND R. KHAN where 0 < q < 1. It can easily be seen that for n N and z D q { a n z n } = [n,q]a n z n 1, 3 where [n,q]= 1 qn n 1 q = 1 + q l, [0,q]=0. l=1 For any non-negative integer n the q-number shift factorial is defined by { 1, n = 0, [n,q]! = [1,q][2,q][3,q] [n,q], n N. Consequently the q-generalized Pochhammer symbol for x > 0is { 1, n = 0, [x,q] n = [x,q][x + 1,q]...[x + n 1,q], n N, and for x > 0, let the q-gamma function Γ q x + 1=[x,q]Γ q x; Γ q 1=1. Motivated by the previous discussion above and having in mind [8, 10, 13, 17, 20, 22], we define a new subclass K p,q α,δ,a,b of A p as follows. DEFINITION 1. Let 1 B < A 1, 0 α < 1and0< q < 1. Then a function f A p belongs to the class K p,q α,δ,a,b if t p z p+1 q F δ z [p,q]gzgtz p +[pb +p αa B]z, 4 p1 + Bz where gz is in the class of p-valent starlike functions of order 1 2 denoted by S p 1/2 and F δ z= 1 δ[p,q] f z+δz q f z, [p,q] with denotes subordination. We note that 1. For A = 1, B = 1, δ = 0andq 1 we get the class of multivalent close-toconvex functions order α. 2. For A = 1, B = 1, δ = 0, α = 0andq 1 the class of multivalent closeto-convex functions occurs. 3. For A = 1, B = 1, δ = 0, p = 1andq 1 the class of close-to-convex functions of order α is covered.

3 ON q -ANALOGUE OF ANALYTIC CLOSE-TO-CONVEX FUNCTIONS 85 Equivalently, a function f z A p is in the class K p,q α,δ,a,b if and only if t p z p+1 q F δ z [p,q]gzgtz 1 B +1 p α A B B t p z p+1 q F δ z < 1. 5 [p,q]gzgtz For our main results we will need the following Lemma. LEMMA 1. [21] Let hz =1 + d n z n k z =1 + k n z n in D. If kz is univalent in D and k D is convex, then d n k 1, for all n N. 2. The main results and their consequences THEOREM 1. Let g i z S p α i, i = 1,2. Then g 1 t 1 zg 2 t 2 z t p 1 t p 2 zp where γ = α 1 + α 2 1 and 0 < t i 1. S p γ, Proof. Let g i z S p α i.thenbydefinition for all i = 1,2. Now writing Re t izg i t iz pg i t i z > α i, Gz= g 1t 1 zg 2 t 2 z t p 1 t p 2 zp, we obtain that It follows that zg z Gz = t 1zg 1 t 1z + t 2zg 2 t 2z p g 1 t 1 z g 2 t 2 z zg z pgz = t 1zg 1 t 1z pg 1 t 1 z + t 2zg 2 t 2z pg 2 t 2 z 1. Re zg z pgz = Re t 1zg 1 t 1z pg 1 t 1 z + Re t 2zg 2 t 2z pg 2 t 2 z 1 > α 1 + α 2 1 = γ, which implies that Gz Sp γ, which completes the proof. Now for t 1 = 1, t 2 = t and g 1 z=g 2 z=gz we get the following corollary.

4 86 B. AHMAD,M.FAROOQ AND R. KHAN COROLLARY 1. If gz S p 1/2 then Gz= gzgtz t p z p S p 0 := S p. THEOREM 2. Let f A p be of the form 1. Then the function f K p,q α,δ,a,b if and only if the following inequality holds p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p +p1 + A αa Bp + n[p,q] 2 p αa B[p,q] 2. 6 Proof. Suppose that the inequality 6 holds. Then to show that f K p,q α,δ,a,b we only need to prove 5. For this consider H := z qf δ z [p,q]gz 1 B+1 p α A B B z qf δ z [p,q]gz Now with the help of 2, 3,1 and Gz=z p + where p is a positive integer and with Λ n = = k=1 z q F δ z [p,q]gz B+1 p α A B [p,q]gz Bz q F δ z. 7 b k+p z k+p, z D, 8 1 δ[p,q]+δ[p + n,q] [p,q] we get that the expression in 7 equals H = = = [p,q]z p + Λ n[n+p,q]a n+p z n+p [p,q]z p [p,q] b n+pz n+p B+1 p α A B [p,q]z p +[p,q] b n+pz n+p B[p,q]z p + Λ n[n+p,q]a n+p z n+p Λ n[n+p,q]a n+p z n+p [p,q] b n+pz n+p 1 p α A B[p,q]zp BΛ n[n+p,q]a n+p z n+p + B+1 p α A B [p,q] b n+pz n+p Λ n[n+p,q]a n+p z n [p,q] b n+pz n 1 p α A B[p,q] BΛ n[n+p,q]a n+p z n + B+1 p α A B [p,q] b n+pz n Λ n[n+p,q] a n+p +[p,q] b n+p 1 p α A B[p,q] BΛ n[n+p,q] a n+p B+1 p α A B [p,q] b n+p 9 Since gz Sp 1/2, by the Corrolary 1 Gz is in the class S p having representation 8; then b p+n p + n 10 and H < 1,wherewehaveused6 which completes the direct part.

5 ON q-analogue OF ANALYTIC CLOSE-TO-CONVEX FUNCTIONS 87 Conversely, let f K p,q α,δ,a,b be given by 1. Then from 5 we have for z D, that z q F δ z [p,q]gz 1 B +1 p α A B B z qf δ z [p,q]gz = Λ n[n+p,q]a n+p z n [p,q] b n+pz n 1 p α A B[p,q] BΛ n[n+p,q]a n+p z n + B+1 p α A B [p,q] b n+pz n. Since Rez z, wehave { Re Λ n[n+p,q]a n+p z n [p,q] b n+pz n 1 p α A B[p,q] BΛ n[n+p,q]a n+p z n + B+1 p α A B [p,q] b n+pz n z q F δ z [p,q]gz Now choose values of z on the real axis so, that denominator in 11 and letting z 1 through real values, we obtain 6. } < 1 11 is real. Upon clearing the THEOREM 3. Let f K p,q α,δ,a,b and be of the form 1. Then a p+n [p,q] 2 p+n+ p αa Bn 12p+n. [p+n,q]1 δ[p,q]+δ[p+n,q] 2p If and Proof. The f A p belonging to the class K p,q α,δ,a,b satisfies and it is of the form since t p z p+1 q F δ z [p,q]gzgtz 1 +[B +1 p α A B]z. 1 + Bz Gz= gzgtz t p z p hz= z qf δ z [p,q]gz, 12 hz=1 + d n z n, hz 1 +[B +1 p α A B]z = Bz p αa B z +... p Then by Lemma 1 we get d n p αa B 13 p

6 88 B. AHMAD,M.FAROOQ AND R. KHAN Now putting the series expansions of hz,gz and f z in 12, simplify and compare the coefficients of z p+n on both sides 1 δ[p,q]+δ[p+n,q] [p,q] 2 [p + n,q]a p+n = b p+n +b p+n 1 d 1 +b p+n 2 d b p+1 d n 1. Taking modulus on both sides, using the triangle s inequality and then by 13 and 10 we obtain 1 δ[p,q]+δ[p + n,q] [p,q] 2 [p + n,q] n 1 ap+n p αa B n + p + p i=2 p + i, which implies that ap+n [p,q] 2 p+n+ p αa Bn 12p+n, [p+n,q]1 δ[p,q]+δ[p+n,q] 2p where a 1 = 1. THEOREM 4. Assume f K p,q α,δ,a,b has the form 1. Then for z = r [p,q]1 Crr p 1 1 Br1 + r 2p q F δ z [p,q]1 +Crr p Br1 r 2p where C = B +1 α p A B. Proof. Suppose that f K p,q α,δ,a,b. Then we can write z q F δ z [p,q]gz 1 +Cz 1 + Bz. Accordingly with z = r z q F δ z [p,q]gz 1 CBr2 1 B 2 r 2 routine simplificationsgive us 1 Cr 1 Br z q F δ z [p,q]gz Because Gz S p, thus C Br 1 B 2 r 2, 1 +Cr 1 + Br. 14 r p 1 + r 2p Gz r p. 15 2p 1 r Now by replacing 15 in 14 we obtain the required result.

7 ON q -ANALOGUE OF ANALYTIC CLOSE-TO-CONVEX FUNCTIONS 89 THEOREM 5. Let f K p,q α,δ,a,b has the form 1. Then for z = r r p 1 τ 1 f z r p 1 + τ 1, where τ 1 = [p,q]2 p αa B p1 + A αa Bp + 1. p1 + B[1 + p,q]1 δ[p,q]+δ[p + 1,q] Proof. Consider f z = zp + n+p z a n+p z p + = r p + a n+p z n+p an+p r n+p. As z = r < 1, so r n+p < r p and f z r 1 p + an+p 16 Similarly f z r p 1 an+p 17 Since 6, this implies that But p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p +p1 + A αa Bp + n[p,q] 2 p αa B[p,q] 2. p1 + A αa Bp + 1[p,q] 2 + p1 + B[1 + p,q]1 δ[p,q] +δ[p + 1,q] a n+p p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p +p1 + A αa Bp + n[p,q] 2 p αa B[p,q] 2,

8 90 B. AHMAD,M.FAROOQ AND R. KHAN which gives an+p [p,q] 2 p αa B p1+a αa Bp+1 p1+b[1+p,q]1 δ[p,q]+δ[p+1,q] By putting the right hand side expression value in 16 and 17 we conclude the assertion. Here, and in what follows, we denote by C p β the class of p-valent convex functions of order β where 0 β < 1. THEOREM 6. Let f K p,q α,δ,a,b. Then f C p α for z < r 1,where p 2 1 p β 1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] n r 1 =, p + nn + p β p αa B p1 + A αa Bp + 1 [p,q] 2 and n N. Proof. Let f K p,q α,δ,a,b. To prove that f C p β we only have to show: zf z p 1 f z zf z+1 2β + p f z < 1. Using 1 along with some simplifications From 6, we obtain that p + nn + p β an+p z n < pp β p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p p αa B p1 + A αa Bp + 1[p,q] 2 p1+b[n+p,q]1 δ[p,q]+δ[p+n,q] p αa B p1+a αa Bp+1[p,q] 2 an+p < 1. The inequality 18 will be satisfied if the following holds < which implies that z n < p + nn + p β an+p z n pp β p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] p αa B p1 + A αa Bp + 1[p,q] 2 a n+p, p 2 p β 1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] p + nn + p β p αa B p1 + A αa Bp + 1 [p,q] 2,

9 and so ON q -ANALOGUE OF ANALYTIC CLOSE-TO-CONVEX FUNCTIONS 91 p 2 1 p β 1+B[n+p,q]1 δ[p,q]+δ[p+n,q] n z < = r1 p+nn+p βp αa B p1+a αa Bp+1[p,q] 2. We denote by S p β the class of p-valent Starlike functions of order β where 0 β < 1. THEOREM 7. Let f K p,q α,δ,a,b. Then f S p β for z < r 2,where 1 p β p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] n r 2 =, n + p β p αa B p1 + A αa Bp + 1[p,q] 2 and n N. Proof. We know that f Sp β if and only if zf z pfz zf z+p 2β f z 1. Using 1 upon reducing the material we conclude Now from 6 we obtain that n + p β p β an+p z n < p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p < 1. p αa B p1 + A αa Bp + 1[p,q] 2 Inequality 19 is valid if < This gives z n < n + p β p β a n+p z n p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] an+p. p αa B p1 + A αa Bp + 1[p,q] 2 p β p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] n + p β p αa B p1 + A αa Bp + 1 [p,q] 2,

10 92 B. AHMAD,M.FAROOQ AND R. KHAN and hence 1 p β p1 + B[n + p,q]1 δ[p,q]+δ[p + n,q] n z < = r2 n + p β p αa B p1 + A αa Bp + 1[p,q] 2, Thus we obtain the required result. REFERENCES [1] I. ALDAWISH AND M. DARUS, Starlikeness of q-differential operator involving quantum calculus, Korean J. Math. 22 4, [2] H. ALDWEBY AND M. DARUS, Some subordination results on q-analogue of Ruscheweyh differential operator, Abstr. Appl. Anal., Vol. 2014, Article ID , 6 pages [3] A. ARAL, On the generalized Picard and Gauss Weierstrass singular integrals, J. Comput. Anal. Appl. 8 3, [4] A. ARAL, V. GUPTA AND R. P. AGARWAL, Applications of q-calculus in Operator Theory, Springer-Verlag New York, [5] A. ARAL AND V. GUPTA, Generalized q-baskakov operators, Math. Slovaca 61 4, [6] A. ARAL AND V. GUPTA, On q-baskakov type operators, Demonstr. Math. 42 1, [7] G. A. ANASTASSIUAND S. G. GAL, Geometric and approximation properties of generalized singular integrals, J.KoreanMath.Soci.23 2, [8] J. DZIOK, G. MURUGUSUNDARAMOORTHY AND J. SOKOŁ, On certain class of meromorphic functions with positive coefcients, Acta Math. Sci. Ser. B 32 4, [9] M. R. GANIGI AND B. A. URALEGADDI, New criteria for meromorphic univalent functions, Bull. Math. Soc. Sci. Math. Roumanie N.S., 33 81, [10] A. HUDA AND M. DARUS, Integral operator defined by q-analogue of Liu-Srivastava operator, Studia Universitatis Babes-Bolyai Series Mathematica 58 4, [11] F. H. JACKSON, On q-definite integrals, The Quarterly Journal of Pure and Applied Mathematics 41, [12] F. H. JACKSON, On q-functions and a certain difference operator, Earth and Environmental Science Transactions of The Royal Society of Edinburgh 46 2, [13] W. JANOWSKI, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28, [14] S. KANAS AND D. RĂDUCANU, Some class of analytic functions related to conic domains, Math. Slovaca. 64 5, [15] S. MAHMMOD AND J. SOKÓŁ, New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Results Math. 71 4, [16] A. MOHAMMED AND M. DARUS, A generalized operator involving the q-hypergeometric function, Mat. vesn. 65 4, [17] T. M. SEOUDY AND M. K. AOUF, Coefficient estimates of new classes of q-starlike and q-convex functions of complex order, J. Math. Inequal. 10 1, [18] H. M. SRIVASTAVA AND D. BANSAL, Close-to-convexity of a certain family of q-mittag-leffler functions, J. Nonlinear Var. Anal. 1 1, [19] H. M. SRIVASTAVA, Univalent functions, fractional calculus and associated generalized hypergeometric functions, in Univalent Functions, Fractional Calculus, and Their Applications H. M. Srivastava and S. Owa, Editors, Halsted Press Ellis Horwood Limited, Chichester, pp , John Wiley and Sons, New York, Chichester, Brisbane and Toronto, [20] C. POMMERENKE, On meromorphic starlike functions, Pacific J. Math. 13, [21] W. ROGOSINSK, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 2,

11 ON q -ANALOGUE OF ANALYTIC CLOSE-TO-CONVEX FUNCTIONS 93 [22] B. A. URALEGADDI AND C. SOMANATHA, Certain diferential operators for meromorphic functions, Houston J. Math. 17 2, , [23] W. C. ROYSTER, Meromorphic starlike multivalent functions, Trans. Amer. Math. Soc , Received December 1, 2017 Bakhtiar Ahmad Abdul Wali Khan University Mardan Mardan, Pakistan pirbakhtiarbacha@gmail.com Muhammad Farooq Abdul Wali Khan University Mardan Mardan, Pakistan farooq4math@gmail.com Raees Khan Abdul Wali Khan University Mardan Mardan, Pakistan raeeskhatim@gmail.com Journal of Classical Analysis jca@ele-math.com