Inclusion properties of Generalized Integral Transform using Duality Techniques

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1 DOI.763/s y Moroccan J. Pure and Appl. Anal.MJPAA Volume 22, 26, Pages 9 6 ISSN: RESEARCH ARTICLE Inclusion properties of Generalied Integral Transform using Duality Techniques Satwanti Devi a and A. Swaminathanb Abstract. Let Wβ α, γ be the class of normalied analytic functions f defined in the region < and satisfying Re eiφ h i α+2γf / + α 3γ + γ / f /f + / + f /f f / f /f β >, with the conditions α, β <, Rγ, > and φ R. For a non-negative and real valued integrable function λt with λtdt =, the generalied non-linear integral transform is defined as Z / Vλ f = λt f t/t dt. The main aim of the present work is to find conditions on the related parameters such that Vλ f Wβ α, γ, whenever f Wβ22 α2, γ2. Further, several interesting applications for specific choices of λt are discussed. 2 Mathematics Subject Classification. 3C45, 3C55, 32C5, 32C37, 33C2. Key words and phrases. Integral Transforms, Analytic functions, Hypergeometric functions, Convolution, Duality techniques. Received February 5, 26 - Accepted June 6, 26. c The Authors 26. This article is published with open access by Sidi Mohamed Ben Abdallah University a Department of Applied Sciences The NorthCap University, Sector 23-A Gurgaon, Haryana, India. ssatwanti@gmail.com, satwantidevi@ncuindia.edu b Department of Mathematics Indian Institute of Technology, Roorkee, Uttarkhand, India. swamifma@iitr.ac.in, mathswami@gmail.com. 9

2 92 S. DEVI AND A. SWAMINATHAN. Introduction Let A be the class of all normalied and analytic functions f defined in the open unit disk D = { C : < } that have the Taylor series representation f = + a n n. Let λt :, ] R be a non-negative integrable function which satisfies the condition λtdt =. For f A, consider the generalied integral transform defined by / ft F f := Vλ f = λt dt, > and D. t The power appearing in and elsewhere in this manuscript are meant as principal values. We are interested in the following problem. Problem.. Find a class of admissible functions f A that are carried by the integral operator defined by to a class of analytic functions. Consideration of such problem for various type of non-linear integral operators exist in the literature. For example, the integral operator ] c + α /α I g,c,α f = f α tg c tg tdt g c where g A, g =, g and g for D\{}, was considered by T. Bulboacă in 6, P.58] for analying various inclusion properties involving interesting classes of analytic functions. Choosing g = gives ] c + α /α I c,α f = f α tt dt c. 2 c This operator has a rich literature and was considered among several authors by S. S.Miller and P.T. Mocanu 2, P.39] see also 9, P.228] for studying various inclusion properties. Even though particular values of the operator given by 2 may be related to the operator under some restrictions, these two operators are entirely different. But the existing literature related to the operator 2 motivate us to consider suitable classes of analytic functions that can be studied with reference to the operator given by. For this purpose, we define the class Wβ α, γ in the following way. { f Wβ α, γ:= f A : Re e α+2γ + iφ α 3γ +γ f + ] f + f f f } f β >, D, φ R. f Note that, for the particular case W β α, P α, β, the operator was examined by R. Aghalary et al. ] using duality techniques. Further, for the case =, this operator reduces to the one introduced by R. Fournier and S. Ruscheweyh 6], n=2

3 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM 93 that contains some of the well-known operator such as Bernardi, Komatu and Hohlov as its special cases for particular choices of λt, which has been extensively studied by various authors for details see 2, 3, 4, 9,, ] and references therein. It is also interesting to note that results related to other particular cases namely, Wβ α, γ W βα, γ, considered by R.M. Ali et al 2] and Wβ α++α, α R α, β exist in the literature. Note that F := f/ R α, β F P α, β. Hence the class R α, β is closely related to the class P α, β. Inclusion properties of Vλ f W β α, γ for subclasses of analytic functions that have geometrical meaning were considered by the authors of this work in 2, 3]. Since it is challenging to solve Problem. completely, in this manuscript the following particular cases are addressed. Problem.2. For given ξ <, obtain the sharp bounds for β such that i Vλ f W ξ, whenever f W β α, γ and ii Vλ f W ξ α, γ whenever f W β α, γ. Main results related to Problem.2 and its consequences are given in Section 2 whereas the required proofs are given in Section 3 separately to provide the readers a collective view of the results. 2. Main results and their consequences The following Lemma is used in obtaining the main results. Lemma ] Let β, β 2 < and φ R. Then for the analytic functions p and q defined in the region D with p = = q, along with the conditions Re e iφ p β > and Req β 2 > implies Re e iφ p q β >, where β = 2 β β 2. Here denotes the convolution or Hadamard product of two normalied analytic functions f = + n=2 a n n and f 2 = + n=2 b n n, defined in D, given by f f 2 = + a n b n n. The parameters µ, introduced in 2] are used for further analysis that are defined by the following relations Clearly 3 leads to two cases. i γ = = µ =, = α. ii γ > = µ >, >. n=2 µ = γ and µ + = α γ. 3 Following theorem addresses the first question of Problem.2 and the proof of the same is given in Section 3.

4 94 S. DEVI AND A. SWAMINATHAN Theorem 2.. Let γ µ, and >. Further let ξ < and β < be defined by the relation ξ ds λt 2 +ts µ/ + ] drds dt +tr / s µ/ γ >, β = ξ λt 2 α + t + ] dr dt α +tr α/ γ =. 4 Then for f Wβ α, γ, the function Vλ f W ξ,. The value of β is sharp. Remark 2.. For γ =, Theorem 2. reduces to, Theorem 2.]. 2 For =, Theorem 2. gives the result of 23, Theorem 2.]. The integral operator defined by the weight function λt = + ct c, c >, 5 is known as generalied Bernardi operator denoted by Bc. This operator is the particular case of the generalied integral operators, considered in the work of R. Aghalary et al. ] that follows in the sequel. The operator corresponding to the value =, was introduced by S. D. Bernardi 7]. Now, using the integral operator Bc, the following corollary is stated as under. Corollary 2.. Let γ µ,, > and c >. Further let ξ < and β < be defined by the relation, 2 + c, + µ ; ξ2 + c + µ 2 + c 3 F c, 2 + µ +, 2 + c, + 4F 3 µ c, 2 +, β = µ ξ2 + c 2 + c + + α α, 2 + c α 2 F 3 + c 3F 2 Then for f Wβ α, γ, the function + c Bcf = c + ;, + ; 2 +, 2 + c, + α 3 + c, 2 + ; α belongs to Wξ,. The value of β is sharp. ω c fω dω /,, γ >,, γ =.

5 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM 95 Here p F q c,..., c p ; d,..., d q ; or p F q denotes the generalied hypergeometric function given by c,..., c p c n..., c p n pf q ; = d,..., d q d n..., d q n n! n, D, 6 n= where c i i =,,..., p and d j j =,,..., q are the complex parameters with d j,,... and p q +. In particular, 2 F is the well-known Gaussian hypergeometric function. For any natural number n, the Pochhammer symbol or shifted factorial ε n is defined as ε = and ε n = εε + n. The generalied hypergeometric series pf q defined in 6, converges absolutely for all in < if p < q +, and for D if p = q +. The following result gives conditions such that Vλ f W ξ α, γ whenever f α, γ which addresses the second part of Problem.2. W β Theorem 2.2. Let γ µ, and >. Further let ξ < and β < be defined by the relation β +ξ t β = ξ λt dt. 7 + t Then for f Wβ α, γ, the function Vλ f W ξ α, γ. The value of β is sharp. It is interesting to note that Theorem 2.2 cannot be reduced to Theorem 2. for particular values of the parameters. However, for particular values of the parameters given in Theorem 2.2 several results exist in the literature and few of them are listed below. Remark 2.2. For γ =, =, α= Theorem 2.2 reduces to 6, Theorem 2]. 2 For γ = and =, Theorem 2.2 gives the result of 7, Theorem 2.6]. 3 For γ =, Theorem 2.2 reduces to, Theorem 2.2]. 4 For =, Theorem 2.2 gives the result of 23, Theorem 2.3]. On setting λt given in 5, Theorem 2.2 will lead to the following result. Corollary 2.2. Let γ µ,, > and c >. Further let ξ < and β < be defined by the relation β = 2 + c 2F, 2 + c; 3 + c; 2 + c ξ. 2 + c 2 F, 2 + c; 3 + c; Then for f Wβ α, γ, the function B cf Wξ α, γ. The value of β is sharp. Let λt = Γc c a, a ΓaΓbΓc a b + tb t c a b 2F c a b + ; t, then the integral operator defined by the above weight function λt is the known as generalied Hohlov operator denoted by Ha,b,c. This integral operator was considered in the work of A. Ebadian et al. 5]. Its representation in the form of convolution

6 96 S. DEVI AND A. SWAMINATHAN is given as H := Ha,b,c f = 2F a, b; c; f. For the case =, the convolution form of the reduced integral transform is given in the work of Y. C. Kim and F. Ronning 7] and studied by several authors later. The operator Ha,b,c with a = is the generalied Carlson-Shaffer operator L b,c 8]. Corresponding to these operators the following results are obtained. { } Theorem 2.3. Let α > γ + 2a+, >, < a min, and 2α γ < + b < c a < 2. Then for f Wβ,, the function Ha,b,c f belongs to the class Wβ α, γ, where β = 2 β β 2 with β 2 := a α γ + a ] 2F a, b; c; + a ] 2a + α γ + 2F a +, b; c; aa + γ + 2 2F a + 2, b; c;. The value of β is sharp. For a =, Theorem 2.3 lead to the following result. Corollary 2.3. Let α > γ + 3, >, and < b < c <. Then for f Wβ,, the function L b,c f belongs to the class W β α, γ, where β = 2 β β 2 with β 2 := α γ + ] 2F, b; c; + α γ + 3 ] 2F 2, b; c; + 2γ 2 2 F 3, b; c;. The value of β is sharp. It is important to note that similar results for various operators considered in 2, 3] by the authors of this work are already obtained and are appearing elsewhere. 3. Proofs of theorems 2., 2.2 and 2.3 Proof of Theorem 2.. Since the case γ = µ =, = α > corresponds to, Theorem 2.], it is enough to obtain the condition for γ >. Let f H := α+2γ + α 3γ + γ f + ] f f f + f f. f 8 f Now, considering P = a simple computation gives f f P + P =. 9 f

7 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM 97 Now, differentiating 9 twice and applying successively in 8 leads to α γ H = + γ α γ P + 2γ P + γ 2 2 P 2 which upon using 3 leads to µ + H = + µ µ + P + 2 2µ P + µ 2 P 2. Considering a series expansion of the form P = + n= b n n in the above equality gives H = + 2 n= + nµ + nb n n. From 9 and, it is a simple exercise to see that f f n + = + f + nµ + n n H. Since + n= n + + nµ + n n = is equivalent to f f = f + + n= 3F 2 2,, µ ; 3F 2 2,, µ ; +, + ; µ 3F 2,, µ ; +, + ] ;, µ +, + µ 3F 2,, µ ; +, ; + µ Taking the logarithmic derivative on both sides of we have F F F = Now, substituting 2 in 3 will give F F F λt = t dt + λt t dt 3F 2 f f 3F 2,, µ ; + ] ; H. 2 f. 3 2,, µ ; +, + µ, + ; µ ] H. ;

8 98 S. DEVI AND A. SWAMINATHAN For f Wβ α, γ, then it is easy to see that for some φ R, Re eiφ H β >. Therefore, for γ >, it is required to prove the claim that Re λt 3F 2 2,, µ ; +, + ; t µ + 3F 2,, µ ; +, + ] ; t dt > ξ µ 2 β, 4 which by applying Lemma 2., implies that F Wξ,. Now it is enough to verify inequality 4. From the identity 3F 2 2, a, b; c, d; = c 3 F 2, a, b; c, d; c 2 3 F 2, a, b; c, d;, it follows that 3F 2 2,, µ ; + Therefore Re = Re λt λt, + µ ; = 3F 2 2,, µ ; +, + µ + 3F 2,, µ ; +, 2F, µ ; + µ 2F 3F 2,, µ ; + ; t + µ ] ; t dt, µ ; + µ, + ; µ ;. ; t + 3F 2,, µ ; +, + ] ; t dt. µ Since the integral form of the following generalied hypergeometric function is given as 2F, µ ; + ds ; t = µ ts µ/ and therefore Re λt 3F 2,, µ ; +, + ; t = µ 2F, µ ; + ; t + µ drds tr / s µ/, 3F 2,, µ ; +, + ] ; t dt µ ds = Re λt ts µ/ + It is evident that Re t >, for <. Thus by the given hypothesis +t ds Re λt ts µ/ + drds tr / s µ/ ] drds dt. tr / s µ/ ] dt

9 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM 99 λt = ξ 2 β Here β < if and only if λt ξ 2 ds + ts µ/ + ds +ts + µ/ drds + tr / s µ/ ] dt ] drds dt < +tr / s µ/ Since ξ < and λtdt =, therefore the above condition holds if and only if λt ds +ts ] drds dt > µ/ +tr / s µ/ or λtgtdt >, where Gt = n= n= n t n + nµ 2 n= ] n t n. + nµ + n Since µ, and are positive and < t <, therefore by Leibnit s test both the series n t n n t n and + nµ + nµ + n n t n are convergent with + nµ < and n t n + nµ + n <. 2 n= n= Further, using the previous conditions, it is easy to see that Gt > 2 ] = t,. 2 By the given hypothesis λt is a non-negative integrable function. Hence n= λtgtdt > or β <. This completes the proof. Now, to verify the sharpness let f Wβ α, γ, therefore it satisfies the differential equation µ f 2 /µ /µ /+ / = β + β + 5 with β < defined in 4. From 3, 5 and the series representation of P, an easy calculation gives f f n + n = + 2 β f + n + nµ. n=

10 S. DEVI AND A. SWAMINATHAN Therefore from 3, we have F F = F λt = + 2 β + 2 β n= n= n + τ n n + n + nµ n + t n dt + n + nµ where τ n = tn λtdt. By the simple adjustment, 4 can be written as β = 2 ds λt ξ + ts + µ/ or equivalently, β = 2 ξ As τ n = tn λtdt, or β = 2 ξ λt n= n= ξ = + 2 β drds + tr / s µ/ 6 ] dt n t n + nµ + 2 ] n t n dt. + nµ + n n= n τ n + nµ 2 n τ n + nµ + n n= n= Further using 6 and 7, we have F F = + 2 β F = n n + τ n + nµ + n. 7 n= n n + τ n + nµ + n = ξ, which clearly implies the sharpness of the result. Proof of Theorem 2.2. Since the case γ = µ =, = α > corresponds to, Theorem 2.2], therefore the case γ > is taken into consideration. Now, for the function f Wβ α, γ, let G = H β β. 8 Then it is easy to see that for some φ R, Re e iφ G >, where H is defined in 8. Now, the following two cases are discussed. At first, let γ α/3. From 8, 3 and 8, F F F = λt t dt α 3γ +γ/ β + βg α+2γ f

11 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM γ f f + f f f f. f 9 Now, from and 3, it is a simple exercise to see that F + F F F F F F λt = t dt f + f f f f f. 2 f Therefore 9 and 2 leads to F F ] λt = β + β F t dt α 3γ + γ/ G F α + 2γ + γ α 3γ + γ/ + F F F F F or equivalently, β + β + α 3γ +γ λt t dt F G = α+2γ F + F + ] F F F F F F. F 2 Now consider γ = α/3. It is easy to see that the equation 8 is equivalent to γ + γ f + γ f + f f f f f f = β + βg. Therefore the above expression along with 3 and 2 gives λt G β + β t dt = γ + γ F +γ F + F F F F F, F which coincides with 2, when γ = α/3. Moreover, F Wξ α, γ if, and only if, J W α, γ where the functions F and J are defined by the relation J = F ξ ξ F J = ξ + ξ. 22

12 2 S. DEVI AND A. SWAMINATHAN From the above expression, a simple computation give F F J J = ξ + ξ. 23 F J Further using 23 will give F + F F F F F F = ξ J + J J J + ξ. J J J 24 Putting the values from 22, 23 and 24 in 8, it follows that J G = α + 2γ + α 3γ + γ J + + J ] J J J J, J which implies that the function J Wα, γ. Further using 2 will give β ξ ξ + β λt J ξ t dt G = α+2γ + α 3γ + γ J + + J ] J J J J. J As Re e iφ G >, from the result given in 2, P. 23] and by the above expression, it is easy to note that J α+2γ + α 3γ +γ J + + J ] J J J J, J if, and only if, β ξ Re ξ + β λt ξ t dt > 2. > Now, using Re Re t β ξ ξ + β ξ Since the expression 7 can be rewritten as the above inequality gives β ξ Re ξ + β ξ +t, <, in the above inequality, it follows that λt t dt > β ξ ξ + β λt ξ + t dt. β ξ β + and this completes the proof. λt ξ dt = + t 2 β, λt t dt > β ξ ξ + β λt ξ + t dt = 2

13 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM 3 Now, to verify the sharpness let f Wβ α, γ. Using the series representation of f in will give F = + 2 β n= where τ n = tn λtdt. Now, consider F H := α + 2γ + α 3γ + γ F + + F Therefore, from 25, a simple calculation gives Now rewriting 7 as H = + 2 β β = 2 ξ 2 τ n n + n + nµ, 25 ] F F F τ n n. n= n τ n. n= F F and finding H at = gives H = ξ, which clearly implies the sharpness of the result. Proof of Theorem 2.3. Since H = 2F a, b; c; f, it is a simple exercise to note that H H f = N, H where N := 2 F a, b; c; + ab c 2F a +, b + ; c + ;. Further from the above expression, it follows that H + ] H H H f = N H H 2, H where N 2 := 2F a, b; c; + 2ab c 2F a +, b + ; c + ; aa + bb F cc + 2 a + 2, b + 2; c + 2;. Therefore α 3γ + γ f + ] f f f + f f f f f + α + 2γ = N 3,

14 4 S. DEVI AND A. SWAMINATHAN where N 3 := 2 F a, b; c; + ab c + α γ + γ 2 F a +, b + ; c + ; aa + bb + γ 2 2F cc + 2 a + 2, b + 2; c + 2;. From the contiguous relation for Gaussian hypergeometric functions 5, Page 96] or the verification can be made by comparing the coefficient of n both the sides it follows that N 3 = a b 2 F a +, b + ; c + ; = c 2 F a +, b; c; 2 F a, b; c;, α γ + a ] 2F a, b; c; + a α γ + ] 2a + 2F a +, b; c; aa + γ + 2 2F a + 2, b; c;. For a >, b > and c >, the integral representation of Gaussian hypergeometric function is given as 7] 2F a, b; c; = Γc ΓaΓbΓc a b+ In view of the above integral form, it is easy to see that N 3 = where N 4 t := a α γ α γ where + a Γc ΓaΓbΓc a b + + a ] 2a + t b t c a b c a, a 2F c a b + ; t t dt. t b t c a b 2 N 4t t dt, t 2 2F c a, a; c a b + ; t c a bc a b ] t 2 F c a, a; c a b; t ac a b + γ 2 2 F c a 2, a + ; c a b ; t =e + e 2 t + n= e 3 c a n a n t n+2, c a b n+ 3 n e := γ 2 e 2 := α γ + 2a + γa + c a 2 2 c a b e 3 := n 2 2 aα + aγ + a 2 γ 2 2 c a bc a b ] + n 3 2 aα + aγ + a 2 γ aα γ + 2a + c a

15 INCLUSION PROPERTIES OF GENERALIZED INTEGRAL TRANSFORM aα + aγ + a 2 γ ac a ] γa + c a 2, 2α γ + 2a + { }, 2α γ which is non-negative when < + b < c a < 2, < a min and α > γ + 2a+. Since Re t >, for <, which on using N +t 3 gives Re N 3 > N 3. Applying Lemma 2. will give the required result. Now, to verify the sharpness, consider the function f = β + β + and N 3 = β 2 + β 2 +. Since β = 2 β β 2, therefore f N 3 = + 4 β β 2 + which clearly implies the sharpness of the result. References = β + β + ] R. Aghalary, A. Ebadian and S. Shams, Geometric properties of some linear operators defined by convolution, Tamkang J. Math , no. 4, ] R. M. Ali, A. O. Badghaish, V. Ravichandran and A. Swaminathan, Starlikeness of integral transforms and duality, J. Math. Anal. Appl , no. 2, ] R. M. Ali, M. M. Nargesi and V. Ravichandran, Convexity of integral transforms and duality, Complex Var. Elliptic Equ , no., ] R. M. Ali and V. Singh, Convexity and starlikeness of functions defined by a class of integral operators, Complex Variables Theory Appl , no. 4, ] G. E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 7, Cambridge Univ. Press, Cambridge, ] T. Bulboacă, Differential Subordinations and Superordinations. Recent Results, House of Scientific Book Publications, Cluj-Napoca, 25. 7] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc , ] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal , no. 4, ] J. H. Choi, Y. C. Kim and M. Saigo, Geometric properties of convolution operators defined by Gaussian hypergeometric functions, Integral Transforms Spec. Funct. 3 22, no. 2, 7 3. ] S. Devi and A. Swaminathan, Integral transforms of functions to be in a class of analytic functions using duality techniques, J. Complex Anal. 24, Art. ID 47369, pp. ] S. Devi and A. Swaminathan, Order of Starlikeness and Convexity of certain integral transforms using duality techniques, 6 pages, Submitted for publication, 2] S. Devi and A. Swaminathan, Starlikeness of the Generalied Integral Transform using Duality Techniques, 24 pages, Submitted for publication, 3] S. Devi and A. Swaminathan, Convexity of the Generalied Integral Transform using Duality Techniques, Submitted for publication. 4] S. S. Ding, Y. Ling and G. J. Bao, Some properties of a class of analytic functions, J. Math. Anal. Appl , no., 7 8.

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