Research Article Some Subordination Results on q-analogue of Ruscheweyh Differential Operator

Transcription

1 Abstract and Applied Analysis Volume 24, Article ID , 6 pages Research Article Some Subordination Results on -Analogue of Ruscheweyh Differential Operator Huda Aldweby and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 436 Bangi, Selangor (Darul Ehsan, Malaysia Correspondence should be addressed to Maslina Darus; maslina@ukm.my Received 5 January 24; Revised 24 March 24; Accepted 25 March 24; Published 4 April 24 Academic Editor: Sergei V. Pereveryev Copyright 24 H. Aldweby and M. Darus. 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. We derive the -analogue of the well-known Ruscheweyh differential operator using the concept of -derivative. Here, we investigate several interesting properties of this -operator by making use of the method of differential subordination.. Introduction Recently, the area of -analysis has attracted the serious attention of researchers. This great interest is due to its application in various branches of mathematics and physics. The application of -calculus was initiated by Jackson [, 2]. He was the first to develop -integral and -derivative in a systematic way. Later, from the 8s, geometrical interpretation of -analysis has been recognied through studies on uantum groups. It also suggests a relation between integrable systems and -analysis. In ([3 5] the -analogue of Baskakov Durrmeyer operator has been proposed, which is based on -analogue of beta function. Another important -generaliation of complex operators is -Picard and -Gauss-Weierstrass singular integral operators discussed in [6 8]. The authors studied approximation and geometric properties of these -operators in some subclasses of analytic functions in compact disk. Very recently, other -analogues of differential operators have been introduced in [9]; see also ([, ]. These -operators are defined by using convolution of normalied analytic functions and -hypergeometric functions, where several interesting results are obtained. From this point, it is expected that deriving -analogues of operators defined on the space of analytic functions would be important in future. A comprehensive study on applications of -analysis in operator theory may be found in [2]. We provide some notations and concepts of -calculus used in this paper. All the results can be found in [2 4]. For n N,<<, we define [n] = n, [n]!={ [n] [n ] [], n=,2,...;, n =. As,[n] n, and this is the bookmark of a analogue: the limit as recovers the classical object. For complex parameters a, b, c, (c C \ {,, 2,...}, <, the-analogue of Gauss s hypergeometric function 2 Φ (a,b;c,,is defined by (a, 2Φ (a,b;c,,= k (b, k k, ( U, (2 (, k (c, k k= where (n, k is the -analogue of Pochhammer symbol defined by (n, k, k=; ={ ( n( n( n 2 ( n k, k N. (3 (

2 2 Abstract and Applied Analysis The -derivative of a function h(x is defined by D (h (x = h(x h(x ( x, =, x=, (4 and D (h( = f (.Forafunctionh( = k observe that D (h ( =D ( k = k k = [k] k ; (5 then lim D (h( = lim [k] k =k k =h (, where h ( is the ordinary derivative. Next, we state the class A of all functions of the following form: f ( =+ a k k, (6 which are analytic in the open unit disk U ={ C : < }. Iff and g are analytic functions in U, wesaythatf is subordinate to g; writtenf g,ifthereisafunctionw analytic in U, withw( =, w( <, forall U, such that f( = g(w( for all U.Ifgis univalent, then f g if and only if f( = g( and f(u g(u. For each A and B such that B<A, we define the function h (A, B; = +A, ( U. (7 +B It is well known that h(a, B; for B is the conformal map of the unit disk onto the disk symmetrical withrespecttotherealaxishavingthecenter( AB/( B 2 for B =±andradius (A B/( B 2.Theboundarycircle cuts the real axis at the points ( A/( B and (+A/(+B. Definition. Let f A. DenotebyR λ the -analogue of Ruscheweyh operator defined by R λ f ( =+ [k+λ ]! [λ]![k ]! a k k, (8 where [a] and [a]! are defined in (. From the definition we observe that, if,wehave lim Rλ f ( =+lim [ [k+λ ]! [λ]![k ]! a k k ] (k+λ! =+ (λ! (k! a k k = R λ f (, where R λ is Ruscheweyh differential operator which was definedin[5] and has been studied by many authors, for example [6 8]. It can also be shown that this -operator is hypergeometric in nature as (9 R λ f ( = 2Φ ( λ+,,,; f(, ( where 2 Φ is the -analogue of Gauss hypergeometric function defined in (2, and the symbol ( stands for the Hadamard product (or convolution. The following identity is easily verified for the operator R λ : λ (D (R λ f ( = [λ+] R λ+ f ( [λ] R λ f (. ( 2. Main Results Before we obtain our results, we state some known lemmas. Let P(β be the class of functions of the form φ ( =+c +c 2 2 +, (2 which are analytic in U and satisfy the following ineuality: Re (φ ( >β, ( β<; U. (3 Lemma 2 (see [9]. Let φ j P(β j be given by (2, where ( β j <;j=,2;then (φ φ 2 P(β 3, (β 3 = 2( β ( β 2, (4 and the bound β 3 is the best possible. Lemma 3 (see [2]. Let the function φ, givenby(2, bein the class P(β.Then Re φ ( >2β + 2( β, ( β <. (5 + Lemma 4 (see [2]. The function ( γ e γ log(,γ=,is univalent in U if and only if γ is either in the closed disk γ or in the closed disk γ +. We now generalie the lemmas introduced in [22] and [23], respectively, using -derivative. Lemma 5. Let h( be analytic and convex univalent in U and h( = and let g( = + b +b be analytic in U.If g ( + D (g ( c Then, for Re(c, h(, ( U;c=, (6 g ( c c t c h (t dt. (7 Proof. Suppose that h is analytic and convex univalent in U and g is analytic in U.Letting in (6, we have g ( + g ( c h(, ( U;c=. (8 Then, from Lemma in [22], we obtain g ( c c t c h (t dt. (9

3 Abstract and Applied Analysis 3 Lemma 6. Let ( be univalent in U and let θ(w and φ(w be analytic in a domain D containing (U with φ(w =when w (U.SetQ( = D ((φ((, h( = θ((+q( and suppose that ( Q( is starlike univalent in U; (2 Re(D (h(/q( = Re((D (θ(/φ(( + (D (Q(/Q( > ( U. If p( is analytic in U,withp( = (, p(u D,and θ(p(+d (p (φ(p( θ((+d ( (φ(( =h(, then p( ( and ( is the best dominant. The proof is similar to the proof of Lemma 5. (2 Theorem 7. Let >, α>,and B<A.Iff A satisfies then ( α Rλ f ( Re (( Rλ f ( >( [λ + ] λ α The result is sharp. Proof. Let +α Rλ+ f ( h(a, B;, (2 u ([λ+] / λ α ( Au Bu du (n. (22 g ( = Rλ f (, (23 for f A.Thenthefunctiong( = + b + is analytic in U. By using logarithmic -differentiation on both sides of (23 and multiplying by,wehave D (g ( g ( by making use of identity (, we obtain = Rλ f ( ; (24 R λ f ( From (, (23, and (26, we get g ( + λ α D [λ+] (g ( h(a, B;. (27 Now, applying Lemma5,wehave g ( [λ+] λ α [λ+] / λα or by the concept of subordination R λ f ( = [λ+] λ α t ([λ+] / λ α ( +At +Bt dt, (28 u ([λ+] / λ α ( +Auw( +Buw( du. (29 In view of B<A and λ>, it follows from (29 that Re ( Rλ f ( > [λ+] λ α u ([λ+] / λ α ( Au Bu du, (3 with the aid of the elementary ineuality Re(w (Re w for Re w > and n.hence,ineuality(22 follows directly from (3. To show the sharpness of (22, we define f Aby R λ f ( = [λ+] λ α For this function, we find that R λ f ( ( α Rλ f ( [λ+] λ α This completes the proof. u ([λ+] / λ α ( +Au +Bu du. (3 +α Rλ+ f ( = +A B, u ([λ+] / λ α ( Au Bu du as. (32 Corollary 8. Let A=2β and B=,where β< and α, λ >.Iff satisfies ( α Rλ f ( +α Rλ+ f ( h(2β, ;, (33 then Re (( Rλ f ( D (g ( g ( = [λ+] λ λ+ R f ( R λf ( [λ]. (25 λ Taking into account that [λ + ] =[λ] + λ,weobtain λ D [λ+] (g ( +g( = Rλ+ f (. (26 >((2β u [λ+] / λ α + 2( β[λ+] λ α u ([λ+] / λ α +u du (n. (34

4 4 Abstract and Applied Analysis Proof. Following the same steps as in the proof of Theorem 7 and considering g( = R λ f(/, the differential subordination (27becomes g ( + λ α D [λ+] (g ( +(2β. (35 + We now assume that ( = φ (w = ( 2γ( ρ[λ+] / λ γ[λ+] w ; λ, θ(w =, (4 Therefore, Re (( Rλ f ( >( [λ+] λ α =( [λ+] λ α u ([λ+] λ / α ( u ([λ+] / λ α =((2β u [λ+] / λ α + 2( β[λ+] λ +( 2βu du +u ((2β + 2( β +u du u ([λ+] / λ α +u du. (36 Theorem 9. Let λ>and ρ<.letγ be a complex number with γ =and satisfy either 2γ( ρ([λ + ] / λ or 2γ( ρ([λ + ] / λ +.Iff Asatisfies the condition then ( Rλ f ( Re ( Rλ+ f ( >ρ, ( U, (37 R λ f ( where ( is the best dominant. Proof. Let γ ( 2γ( ρ([λ+] / λ, ( U, (38 γ p ( =( Rλ f (, ( U. (39 Then, by making use of (, (37, and (39, we obtain + λ D (p ( γ[λ+] p ( +( 2ρ, ( U. (4 then ( is univalent by condition of the theorem and Lemma 4.Further,itiseasytoshowthat(, θ(w,andφ(w satisfy the conditions of Lemma 6.Notethatthefunction Q ( =D ( (φ(( = 2( ρ (42 is univalent starlike in U and h ( =θ((+q( = +( 2ρ. (43 Combining (4 and Lemma 6 we get the assertion of Theorem 9. Theorem. Let α<,λ>and B i <A i.ifeach of the functions f i A satisfies the following subordination condition, ( α Rλ f i ( then, ( α Rλ Θ ( where +α Rλ+ f i ( h(a i,b i ;, (44 +α Rλ+ Θ ( h( 2γ, ;, (45 Θ ( = R λ (f f 2 (, (46 γ= 4(A B (A 2 B 2 ( B ( B 2 ( [λ+] λ α Proof. we define the function h i by h i ( = ( α Rλ f i ( u ([λ+] / λ α +u du. +α Rλ+ f i ( (47 (48 (f i A,i=,2; we have h i ( P(β i,whereβ i =( A i /( B i (i =, 2. By making use of (and(48, we obtain R λ f i ( = [λ+] λ α t ([λ+] / λ α h i (t dt, (i =,2, (49 which, in the light of (46, can show that R λ Θ ( = [λ+] λ α ([λ+] / λ α t ([λ+] / λ α h (t dt, (5

5 Abstract and Applied Analysis 5 where, for convenience, h ( = ( α Rλ Θ ( = [λ+] λ α ([λ+] / λ α +α Rλ+ Θ ( t ([λ+] / λ α (h h 2 (t dt. (5 Note that, by using Lemma 2,wehave(h h 2 P(β 3,where β 3 = 2( β ( β 2. Now, with an application of Lemma 3,wehave Re (h ( = [λ+] λ α u ([λ+] / λ α Re ((h h 2 (udu [λ+] λ α > [λ+] λ α = 4(A B (A 2 B 2 ( B ( B 2 ( [λ+] λ α u ([λ+] / λ α (2β 3 + 2( β 3 +u du u ([λ+] / λ α (2β 3 + 2( β 3 +u du u ([λ+] / λ α +u du = γ, (52 which shows that the desired assertion of Theorem holds. Conflict of Interests The authors declare that they have no competing interests regarding the publication of this paper. Authors Contribution Huda Aldweby and Maslina Darus read and approved the final manuscript. Acknowledgments The work presented here was partially supported by AP-23-9andDIP-23-.Theauthorsalsowouldliketothank therefereesforthecommentsmadetoimprovethispaper. References [] F. H. Jackson, On -definite integrals, The Quarterly Pure and Applied Mathematics,vol.4,pp.93 23,9. [2] F.H.Jackson, On -functions and a certain difference operator, Transactions of the Royal Society of Edinburgh,vol.46,pp , 98. [3] A. Aral and V. Gupta, On -Baskakov type operators, Demonstratio Mathematica,vol.42,no.,pp.9 22,29. [4] A. Aral and V. Gupta, Generalied -Baskakov operators, Mathematica Slovaca,vol.6,no.4,pp ,2. [5] A. Aral and V. Gupta, On the Durrmeyer type modification of the -Baskakov type operators, Nonlinear Analysis: Theory, Methods & Applications,vol.72,no.3-4,pp.7 8,2. [6] G.A.AnastassiouandS.G.Gal, Geometricandapproximation properties of some singular integrals in the unit disk, Ineualities and Applications, Article ID 723, 9 pages, 26. [7] G.A.AnastassiouandS.G.Gal, Geometricandapproximation properties of generalied singular integrals in the unit disk, the Korean Mathematical Society, vol.43,no.2,pp , 26. [8] A. Aral, On the generalied Picard and Gauss Weierstrass singular integrals, Computational Analysis and Applications,vol.8,no.3,pp ,26. [9] A. Mohammed and M. Darus, A generalied operator involving the -hypergeometric function, Matematichki Vesnik, vol. 65,no.4,pp ,23. [] H. Aldweby and M. Darus, A subclass of harmonic univalent functions associated with -analogue of Diok-Srivastava operator, ISRN Mathematical Analysis, vol. 23, Article ID 38232, 6pages,23. [] H. Aldweby and M. Darus, On harmonic meromorphic functions associated with basic hypergeometric functions, The Scientific World Journal, vol.23,articleid64287,7pages, 23. [2] A.Aral,V.Gupta,andR.P.Agarwal,Applications of -Calculus in Operator Theory, Springer, New York, NY, USA, 23. [3] G. Gasper and M. Rahman, Basic Hypergeometric Series,vol.35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 99. [4] H. Exton, -Hypergeometric Functions and Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood,Chichester,UK,983. [5] S. Ruscheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society, vol.49,pp.9 5, 975. [6] M. L. Mogra, Applications of Ruscheweyh derivatives and Hadamard product to analytic functions, International Journal of Mathematics and Mathematical Sciences, vol.22,no.4,pp , 999. [7] K. Inayat Noor and S. Hussain, On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation, Mathematical Analysis and Applications, vol. 34, no. 2, pp , 28. [8] S. L. Shukla and V. Kumar, Univalent functions defined by Ruscheweyh derivatives, International Mathematics and Mathematical Sciences,vol.6,no.3,pp ,983. [9] G. S. Rao and R. Saravanan, Some results concerning best uniform coap- proximation, JournalofIneualitiesinPureand Applied Math,vol.3,no.2,p.24,22. [2] G. S. Rao and K. R. Chandrasekaran, Characteriation of elements of best coapproximation in normed linear spaces, Pure and Applied Mathematical Sciences, vol.26,pp.39 47, 987. [2] M. S. Robertson, Certain classes of starlike functions, The Michigan Mathematical Journal,vol.32,no.2,pp.35 4,985.

6 6 Abstract and Applied Analysis [22] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, The Michigan Mathematical Journal, vol. 28, no. 2, pp , 98. [23] S. S. Miller and P. T. Mocanu, On some classes of firstorder differential subordinations, The Michigan Mathematical Journal, vol. 32, no. 2, pp , 985.

7 Advances in Operations Research Volume 24 Advances in Decision Sciences Volume 24 Applied Mathematics Algebra Volume 24 Probability and Statistics Volume 24 The Scientific World Journal Volume 24 International Differential Euations Volume 24 Volume 24 Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Volume 24 Complex Analysis Volume 24 International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Volume 24 Volume 24 Volume 24 Volume 24 Discrete Mathematics Volume 24 Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis Volume 24 Volume 24 Volume 24 International Stochastic Analysis Optimiation Volume 24 Volume 24