Coefficient Inequalities of a Subclass of Starlike Functions Involving q - Differential Operator

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1 Advances in Analysis, Vol. 3, No., April Coefficient Inequalities of a Subclass of Starlike Functions Involving q - Differential Operator K. R. Karthikeyan 1, K. Amarender Reddy 1* M. Thirucheran 1 Department of Mathematics Statistics, Caledonian College of Engineering, Muscat, Sultanate of Oman. Department of Mathematics, L. N. CollegeAutonomous, Ponneri, Tamilnadu, India. amarenderkommula@gmail.com Abstract We introduce a new subclass of spiralike biunivalent functions involving q-differential operator. We obtain the coefficient estimates Fekete-Szegö inequalities for the functions belonging to this class. Relevant connections with various other known classes have been established. Keywords: Starlike functions, spiralike functions, bi-univalent functions, coefficient inequalities, Fekete-Szegö, symmetric functions. 1 Introduction, Definitions Preliminaries Recently, the area of q analysis has attracted serious attention of researchers. The great interest is due to its applications in various branches of mathematics physics, for example, in the areas of ordinary fractional calculus, optimal control problems, q difference q integral equations in q transform analysis. The generalized q Taylor formula in the fractional q calculus was introduced by Purohit Raina 19]. The application of q calculus was initiated by Jackson 9,10]. He was the first to develop the q integral q derivative in a systematic way. Later, geometrical interpretation of the q analysis has been recognized through studies on quantum groups. Simply, the quantum calculus is ordinary classical calculus without the notion of limits. It defines q calculus h calculus. Here h ostensibly sts for Planck s constant, while q sts for quantum. Mohammed Darus 16] studied approximation geometric properties of these q operators in some subclasses of analytic functions in compact disk. Recently, Purohit Raina 19,0] have used the fractional q calculus operators in investigating certain classes of functions which are analytic in the open disk, Purohit 18] also studied these q operators, defined by using the convolution of normalized analytic functions q hypergeometric functions. A comprehensive study on applications of q calculus in the operator theory may be found in ]. Ramachran et al. 1] have used the fractional q calculus operators in investigating certain bound for q starlike q convex functions with respect to symmetric points. In univalent function theory, all geometrically defined subclasses do have beautiful analytic characterization defined in terms of differential inequality. So extending the existing subclasses in q-calculus has numerous applications. To provide a unified approach to the study of various properties of certain subclasses of A, we introduce a new class of analytic functions of complex order involving q-derivative of f. The q-difference operator denoted as D q fz is defined by D q fz = fz fqz, f A, z U 0, z1 q D q f0 = f 0, where q 0, 1. It can be easily seen that D q fz f z as q 1. Let A denote the class of all analytic functions fz normalized by the condition f0 = f 0 1 = 0 which is of the form fz = z + a n z n, z U. 1 n= If fz is of the form 1, a simple computation yields 1 q n D q fz = q a nz n 1, z U. n=

2 7 Advances in Analysis, Vol. 3, No., April 018 The inverse function of is given by D q gw = qa w 1 + q + q a 3 w q a w +. Let f g be analytic in the open unit disk U. The function f is subordinate to g written as f g in U, if there exists a function w analytic in U with w0 = 0 wz < 1; z U such that fz = gwz, z U. Let S denote the class of all functions in A which are univalent in U. The well known example in this class is the Koebe function, kz defined by z kz = 1 z = z + nz n. Also, let P denote the class of functions of the form n= pz = 1 + c n z n n=1 z U, which are analytic convex in U satisfy the condition Repz > 0; z U. We denote by S, C, K C the familiar subclasses of A consisting of functions which are respectively starlike, convex, close-to-convex quasi-convex in U. Let S α Cα denote the well known subclasses of S which are respectively defined as follows. S α = f A : Re zf z fz > α; 0 α < 1 Cα = f A : Re 1 + z > α; 0 α < 1. f z zf Using Alexer transform, it follows that fz Cα if only if zf z S α. One of the very interesting generalization of the function class S is the so called starlike functions of complex order b which satisfies the condition b zf z fz 1 φz, f A, where φ P, the class of functions with positive real part we denote it by S b φ. Similarly, let C b φ denote the class of functions in A satisfying the condition b zf z fz φz, f A. Note that S b 1 + z/1 z = S b C b 1 + z/1 z = C b are the classes considered by Nasr Aouf in 17] by Wiatrowski in 6]. Our favorite references of the field are 6,7,8] which covers most of the topics in a lucid economical style. It is well known that every function f S has an interval f 1, defined by f 1 fz = z; z U f f 1 w = w ; w < r 0 f; r 0 f 1.

3 Advances in Analysis, Vol. 3, No., April In fact, the inverse function f 1 is given by f 1 w = w a w + a a 3 w 3 5a 3 5a a 3 + a w +. 3 A function f A is said to be biunivalent in U if both fz f 1 z are univalent in U. The Bieberbach conjecture about the coefficient of the univalent functions in the unit disk was formulated by Bieberbach 3] in the year The conjecture states that for every function f S, given by 1, we have a n n for every n. Strict inequality holds for all n unless f is the Koebe function or one of its rotation. For many years, this conjecture remained as a challenge to mathematicians. After the proof of a 3 3 by Löwner in 193, Fekete-Szegö surprised the mathematicians with the complicated inequality µ 3 when µ 1 a 3 µa 1 + exp µ 1 µ when 0 µ 1 3µ 1 when µ 0, which holds good for all values 0 µ 1. Note that this inequality region was thoroughly investigated by Schaefer Spencer ]. Keogh Merkes 1] obtained the following inequalities for the class of convex starlike functions a3 µa 1 max 3, µ 1 a3 µa max 1, 3 µ respectively. For a class functions in A a real or more generally complex number µ, the Fekete-Szegö problem is all about finding the best possible constant Cµ so that a3 µa Cµ for every function in A. Many papers have been devoted to this problem see,5,13,1,15]. Motivated by the concept introduced by Sakaguchi in 3], recently several subclasses of analytic functions with respect to k-symmetric points were introduced studied by various authors. In this paper, we introduce a new subclass of spiralike biunivalent functions using subordination we obtained the estimates of the a a 3 for the functions belonging to this new subclass. Definition 1.1. Let hz be a convex univalent function with h0 = 1. A function f A is said to be in the class Sb λ β, s, t, h if only if it satisfies the analytic condition, e iβ s tz] 1 λ ] D q fz b fsz ftz] 1 λ 1 hz cos β + i sin β e iβ s tw] 1 λ ] D q gw b gsw gtw] 1 λ 1 hw cos β + i sin β, where z U; λ 0; π < β < π ; b C 0. s, t C with s t, t 1. Remark 1.1. On specializing the parameters the function hz, we obtain several new well known subclasses of analytic functions. Here we list a few of them. 1. If we let β = 0 hz = 1 + γ α π i log 1 e πi1 α\γ αz 1 z. then the class Sb λ β, s, t, h reduces to the form α < Re s tz] 1 λ D q fz b fsz ftz] 1 λ 1 < γ. which is analogues to the class introduced by Kuroki Owa in 11].. If we let q 1 b = 1 in Sb λ β, s, t, h. then the class reduces to the class introduced studied by Altınkaya Yalçın in 1].

4 76 Advances in Analysis, Vol. 3, No., April If we set hz = 1+1 αz 1 z, 0 α < 1 in the class Sb λβ, s, t, h, we have Sλ b β, s, t, α defined as Re e iβ s tz] 1 λ f z fsz ftz] 1 λ > α cos β, z U Re e iβ s tw] 1 λ g w gsw gtw] 1 λ > α cos β, where gw = f 1 w, s, t C with s t, t 1. β π Lemma 1.1. Let the function φz given by φz = z U, then h n B 1, n N = 1,, 3,...., π λ 0. B n z n be convex in U. If hz φz Lemma 1.. If pz = 1 + c 1 z + c z + is a function with positive real part in U µ is a complex number,then The result is sharp for the functions given by Main Results n=1 c µc 1 max 1; µ 1. pz = 1 + z 1 + z pz = 1 z 1 z. In this section, we obtain very interesting Fekete-Szegö inequalities for a certain subclass of analytic functions. Theorem.1. Let φz = 1 + B 1 z + B z + with B 1 0. If f A satisfies the differential inequality Then e iβ s tz] 1 λ ] D q fz b fsz ftz] 1 λ 1 φz cos β + i sin β. a3 µa b cos β B 1 B κ 1 κ B 1 b cos β e iβ max 1, + κ 1 B q + λ 1s + t], 5 where κ 1 = 1 + q + q + λ 1s + st + t 1 + q + λ 1s + t]1 λs + t λ1 λ s + t κ = 1 + q + q + λ 1s + st + t µ. The result is sharp. Proof. Let f A satisfy, then there exist Schwarz function w analytic in U with w0 = 0 wz < 1 in U such that e iβ s tz] 1 λ ] D q fz b fsz ftz] 1 λ 1 = φwz cos β + i sin β. 6 Define pz by pz = 1 + wz 1 wz = 1 + c 1z + c z +. 7

5 Advances in Analysis, Vol. 3, No., April Since wz is a Schwarz function, it is clear that Repz > 0 p0 = 1. Therefore pz 1 φz = φ pz = φ c 1 z + = 1 + B 1c 1 z + c c 1 z + c 3 c 1 c + c3 1 B1 c c 1 + B c 1 ] z +. ] z Now by substituting 8 in 6 From this equation, we obtain e iβ s tz] 1 λ D q fz b fsz ftz] 1 λ 1 ] = 1 + B 1c 1 z + B 1 c c 1 + B c 1 ]z + e iβ 1 b λ 1s + t q] a = B 1c 1 cos β + i sin β. cos β e iβ 1 λ 1s + st + t q + q ] λλ 1 a 3 s + t a b +λ 1s + t1 + q + λ 1s + t]a B1 = c c 1 + B c 1 cos β. Or, equivalently e iβ B 1 c 1 b cos β a = 1 + q + λ 1s + t], e iβ B b 1C B1C 1 + BC 1 cos β a 3 = 1 + q + q + λ 1s + st + t On simple computation, we have 1 + q + λ 1s + t]1 λs + t λ1 λ s + t a 1 + q + q + λ 1s + st + t. Therefore where a 3 µa = e iβ b B1C B1C 1 + BC 1 cos β 1 + q + q + λ 1s + st + t 1 + q + λ 1s + t]1 λs + t λ1 λ s + t 1 + q + q + λ 1s + st + t µ e iβ B 1 c 1 b cos β. 1 + q + λ 1s + t] ϑ = 1 a 3 µa = B 1e iβ b cos β κ 1 c ϑc 1 1 B B 1 κ 1 κ B 1 b cos βe iβ 1 + q + λ 1s + t].

6 78 Advances in Analysis, Vol. 3, No., April 018 On rearranging the terms taking modulus both sides, our result now follows by application of Lemma1.. The result is sharp for the functions s tz] 1 λ D q fz fsz ftz] 1 λ = φz s tz] 1 λ D q fz fsz ftz] 1 λ = φz. This completes the proof of the Theorem.1. Corollary.. Let φz = 1+B 1 z +B z +. with B 1 0. If f A satisfies the differential inequality α Re zf ] z b fz 1 < γ. 9 Then a 3 µa β α n1 α 1 cos max 1; B + 1 µbb 1. π β α B 1 The result is sharp. Proof. Let φz = 1 + β α 1 e πi1 α/β α π i log z. 1 z Clearly, it can be seen that φz maps U onto a convex domain conformally is of the form hz = 1 + B n z n where B n nπ i 1 e nπi1 α/β α. From the equivalent subordination condition proved by Kuroki Owa in 11], the inequality 9 can be rewritten in the form zf ] z b fz 1 φz. = β α n=1 Following the steps as in Theorem.1, we get the desired result. Corollary.3. 5] Let φz = 1+B 1 z+b z +. with B 1 0. If f satisfies the following subordination condition ] zdq fz 1 φz b C 0, b fz then a 3 µa B 1b 3] q 1 max 1; B + B 1b 1 3] q 1 B 1 ] q 1 ] q 1 µ. The result is sharp. Proof. The result follows if we let β = 0, λ = 0, t 0 s 1 in Theorem3.1 The result sharp for the function zd q fz = φz zd q fz = φz. fz fz Taking q 1 in the corollary.3, we obtain the Fekete szegö inequality for functions belonging to the class of starlike function of complex order b. Corollary.. See Ravichran et al. ] Let φz = 1 + B 1 z + B z +. with B 1 0. If fz belongs to the class of starlike function of complex order b. Then a 3 µa B 1 b max 1; B + 1 µb 1 b B 1. The result is sharp.

7 Advances in Analysis, Vol. 3, No., April Coefficient Inequalities Of Biunivalent Functions We begin this section with finding the coefficient estimates of Sb λ β, s, t, h. Theorem 3.1. Let fz be of the form 1 suppose that fz is in the class Sb λ β, s, t, h. then b B 1 cos β a λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] λλ 1s + t a 3 b B 1 cos β λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] λλ 1s + t Proof. Let f Sb λ β, s, t, h g denote that inverse of f to U. It follows from the Definition1.1 that there exist functions pz, qz Pthe class of function with positive real part, such that e iβ s tz] 1 λ ] D q fz b fsz ftz] 1 λ 1 = pz cos β + i sin β 10 e iβ s tw] 1 λ D q gw b gsw gtw] 1 λ 1 s, t C with s t, t 1; b C 0; λ 0, β where pz hz qw gw have the forms pz = 1 + p 1 z + p z + qw = 1 + q 1 w + q w +. respectively. It follows from 10 11, we deduce ] = qw cos β + i sin β 11 π, π ] e iβ 1 b λ 1s + t q]a = p 1 cos β, 1 e iβ 1 b λ 1s + st + t q + q ]a 3 λλ 1 s + t a +λ 1s + t1 + q + λ 1s + t]a = p cos β, 13 e iβ 1 b e iβ 1 b λ 1s + t q]a = q 1 cos β, 1 λ 1s + st + t q + q ]a λλ 1 s + t a +λ 1s + t1 + q + λ 1s + t]a λ 1s + st + t q + q ]a 3 = q cos β. 15 From 1 1 we obtain By adding 13 15, we get p 1 = q 1.

8 80 Advances in Analysis, Vol. 3, No., April 018 e iβ 1 λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] b λλ 1s + t a = p + q cos β. 16 Since p, q hu, applying Lemma1.1, we have p m = pm 0 m! q m = qm 0 m! B 1, m N 17 B 1, m N. 18 Applying 17, 18 Lemma1.1 for the coefficients p 1, p, q 1 q, we readily get b B 1 cos β a λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] λλ 1s + t. Subtracting 15 from 13 we have e iβ 1 λ 1s + st + t q + q ]a 3 λ 1s + st + t b +1 + q + q ]a = p q cos β. 19 a 3 = a 3 Or, equivalently e iβ bp + q cos β λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] λλ 1s + t e iβ bp q cos β + λ 1s + st + t q + q ]. Applying 17, 18 Lemma1.1 once again for the coefficients p 1, p, q 1 q, we readily get b B 1 cos β λ 1s + t1 + q + λ 1s + t] + λ 1s + st + t q + q ] λλ 1s + t. This completes the proof of Theroem3.1. Remark 3.1. We note that all the results of Altınkaya Yalçın1] can be obtained if we let q 1 b = 1 in Theorem3.1. Conclusion We have obtained the upper bound for the initial coefficients of a subclass biunivalent functions. Various well-known new results could be obtained as a special case of our results. Since we have defined a class involving q derivative of f, interesting q- analogue of the Sălăgean, Hohlov Dziok-Srivastava operators could be defined used to study various subclasses of the analytic functions. It would be more interesting to extend our study to the class of non-analytic functions. Acknowledgments. The authors thank the referee for valuable comments suggestions on the earlier version of this paper. Particularly, we thank for the comments leading to an improvement in the introduction preliminaries.

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