Research Article On an Integral Transform of a Class of Analytic Functions

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1 Abstract and Applied Analysis Volume 22, Article ID 25954, pages doi:.55/22/25954 Research Article On an Integral Transform of a Class of Analytic Functions Sarika Verma, Sushma Gupta, and Sukhjit Singh Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 486, India Correspondence should be addressed to Sarika Verma, sarika.6984@gmail.com Received 5 June 22; Accepted 4 September 22 Academic Editor: Josip E. Pecaric Copyright q 22 Sarika Verma 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. For α, γ andβ<, let W β α, γ denote the class of all normalied analytic functions f in the open unit disc E : < } such that Re iφ α 2γ f / α 2γ f γf β >, E for some φ R. Itisknown Noshiro 934 and Warschawski 935 that functions in W β, are close-to-convex and hence univalent for β<. For f W β α, γ, we consider the integral transform F V λ f : f t /t dt, whereλ is a nonnegative real-valued integrable function satisfying the condition dt. The aim of present paper is, for given δ<, to find sharp values of β such that i V λ f W δ, whenever f W β α, γ and ii V λ f W δ α, γ whenever f W β α, γ.. Introduction Let A denote the class of analytic functions f defined in the open unit disc E : < } with the normaliations f f, and let S be the subclass of A consisting of functions univalent in E. For any two functions f a n n and g b n n in A, the Hadamard product or convolution of f and g is the function f g defined by ) f g a n b n n.. For f A, Fournier and Ruscheweyh introduced the integral operator F V λ f ) : f t dt, t.2

2 2 Abstract and Applied Analysis where λ is a nonnegative real-valued integrable function satisfying the condition dt. This operator contains some well-known operators such as Libera, Bernardi, and Komatu as its special cases. Fournier and Ruscheweyh applied the famous duality theory to show that for a function f in the class P β ) : f A: φ R Re iφ f β ) } >, E,.3 the linear integral operator V λ f is univalent in E. Since then, this operator has been studied by a number of authors for various choices of. In another remarkable paper, Barnard et al. in 2 obtained conditions such that V λ f P β whenever f is in the class ) } ) P γ β : f A: φ R Re iφ )f γ γf β >, E,.4 with β<, γ. Note that for β<, functions in P β P β satisfy the condition Rf >βin E and thus are close-to-convex in E. A domain D in C is close-to-convex if its compliment in C can be written as union of nonintersecting half lines. In 28, Ponnusamy and Rønning 3 discussed the univalence of V λ f for the functions in the class ) R γ β : f A: φ R Re iφ f γf β ) } >, E..5 In a very recent paper, Ali et al. 4 studied the class W β α, γ ) : f A: φ R Re iφ )f α 2γ α 2γ ) ) } f γf β >, E,.6 where α, γ andβ<. In this paper, they obtained sufficient conditions so that the integral transform V λ f maps normalied analytic functions f W β α, γ into the class of starlike functions. It is evident that W β, P β, W β α, P α β and W β 2γ,γ R γ β. In the present paper, we shall mainly tackle the following problems. For given δ <, find sharp values of β β δ, α such that V λ f W δ, whenever f W β α, γ. 2 For given δ<, find sharp values of β β δ such that V λ f W δ α, γ whenever f W β α, γ. To prove one of our results, we shall need the generalied hypergeometric function p F q,so we define it here.

3 Abstract and Applied Analysis 3 Let α j j, 2,...,p and β j j, 2,...,q be complex numbers with β j /,, 2,... j, 2,...,q. Then the generalied hypergeometric function p F q is defined by pf q p F q α,...,α p ; β,...,β q ; ) α n α p )n β )n β n ) q n! n n p q ),.7 where a n is the Pochhammer symbol, defined in terms of the Gamma function, by a n : Γ a n Γ a, n, a a a n, n N..8 In particular, 2F is called the Gaussian hypergeometric function. We note that the p F q series in.7 converges absolutely for < if p<q andfor E if p q. We shall also need the following lemma. Lemma. see 5. Let β <, β 2 <, and η R. Then, for p, q analytic in E with p q, the conditions Rp >β and Re iη q β 2 > imply Re iη p q δ >,where Main Results We use the notations introduced in 4. Letμ andν satisfy μ ν α γ, μν γ. 2. When γ, then μ is chosen to be, in which case, ν α. When α 2γ, 2. yields μ ν γ μν or μ ν. i For γ>, then choosing μ givesν γ. ii For γ, then μ andν α. Theorem 2.. Let μ, ν satisfy 2.. Further, let δ< be given, and define β β δ, μ, ν by 2 ν ) ) ds ts μ dt ν ) dt 2 α t α ) dη dζ dt} tη ν ζ μ, γ /, ) dη dt} tη α, γ μ, ν α> ). 2.2 If f W β α, γ, thenf V λ f W δ, S. The value of β is sharp. Proof. The case γ μ, ν α> corresponds to Theorem.5 in 2. So we assume that γ>.

4 4 Abstract and Applied Analysis Define α 2γ )f α 2γ ) f γf H. 2.3 Writing f a n n, it follows that H a n nν nμ ) n. n 2.4 It is a simple exercise to see that f H 3 F 2 2, ν, ) μ ; μ ;. 2.5 Let F V λ f, where V λ f is defined by.2. Then for γ /, we can write F f t dt H 3 F 2 2, ν, μ ; μ ; H ) t dt 3 F 2 2, ν, ) μ ; μ ; t dt. 2.6 Since f W β α, γ, it follows that Re iφ H β } > for some φ R. Now, for each γ>, we first claim that R 3 F 2 2, ν, ) μ ; μ ; t dt > 2 ), E, 2.7 which, by Lemma., implies that F W δ,. Therefore, it suffices to verify the inequality 2.7. Using the identity which can be checked by comparing the coefficients of n on both sides 3F 2 2,b,c; d, e; d 3 F 2,b,c; d,e; d 2 3 F 2,b,c; d, e;, 2.8 it follows that 3F 2 2, ν, ) μ ; μ ; ds ν s μ ) dη dζ ν η ν ζ μ. 2.9

5 Abstract and Applied Analysis 5 Thus, 3 F 2 2, ν, ) μ ; μ ; t dt ds ν ts μ ) } dη dζ ν tη ν ζ μ dt. 2. Therefore, for γ>, we have R 3 F 2 2, ν, ) μ ; μ ; t dt > ) ) ds ν ts μ dt ν ) dη dζ tη ν ζ μ dt 2. 2 ), in the view of 2.2. To prove the sharpness, let f W β α, γ be the function determined by α 2γ )f α 2γ ) f γf β ). 2.2 Using a series expansion, we see that we can write f 2 ) nν ν nμ μ )n. 2.3 Then, F V λ f ) 2 ) ψ n nν ν nμ μ )n, 2.4

6 6 Abstract and Applied Analysis where ψ n tn dt. Equation 2.2 can be restated as 2 ν 2 2 ) ) ds ) } ts μ dt ν dη dζ tη ν ζ μ dt ) ds ) } ν ts μ ν dη dζ tη ν ζ μ dt n t n ) ν ) ) ν } dt nμ μ nν ν 2 n nψ n nν ν nμ μ ). 2.5 Finally, F 2 ) nψ n nν ν nμ μ )n, 2.6 which for takes the value F 2 ) n nψ n nν ν nμ μ ) 2 ) } 2 ) δ. 2.7 This shows that the result is sharp. 6. Letting γ andα in Theorem., we obtain the following result of Ruscheweyh Corollary 2.2. Let δ<, and define β β δ, < by β δ } 2 t dt. 2.8 If f W β, P β,thenf V λ f W δ, S. The value of β is sharp. Theorem 2.3. Let δ< and α, γ, and define β β δ < by β δ / t dt. t 2.9 If f W β α, γ, thenv λ f W δ α, γ. The value of β is sharp.

7 Abstract and Applied Analysis 7 Proof. The idea of the proof is similar to the one used to prove Theorem 2 in. Let F V λ f f t /t dt. Clearly, F t dt f. 2.2 Since, f W β α, γ, sowith ) ) ) α 2γ f / α 2γ f γf β g, 2.2 we have R e iφ g >, where φ R. For γ / α/2, f ) ) α 2γ f β g α 2γ α 2γ γ α 2γ f Putting this value in 2.2, F ) t dt ) ) α 2γ f β g α 2γ α 2γ γ α 2γ f Equivalently, F α 2γ g β ) t dt α 2γ F α 2γ γ α 2γ F Thus ) ) α 2γ F / α 2γ F γf g β ) t dt In the case when γ α/2, g f / γf β Since f β ) g γf, 2.27

8 8 Abstract and Applied Analysis This leads to, F γf g β ) t dt, 2.28 which is clearly 2.25 with γ α/2. Further F W δ α, γ if and only if G : F δ / W α, γ. Now using 2.25, weobtain α 2γ )G α γ ) G γg g β δ t dt Since Re iφ g > for some φ R, it follows by duality principle 8, page 23 that α 2γ )G α 2γ ) G γg / 2.3 if, and only if, β δ R t dt > Using R / t > / t,weget β δ R t dt > β δ t dt By using 2.9, we have β δ /2 t dt Thus, β δ t dt 2, 2.34 which implies that β δ R t dt > β δ t dt

9 Abstract and Applied Analysis 9 Thus, we deduce, using duality principle, that α 2γ G / α γ G γg is contained in a half plane not containing the origin. So, G W α, γ and hence F W δ α, γ. To prove the sharpness, let f 2 n / nμ μ nν ν. ) ) n ω n F V λ f 2 ), where ω n nμ μ nν ν t n dt Further, β δ / t dt t 2.37 gives β δ / tdt, t 2.38 or 2 t t dt 2 n ω n Further, assume that H α 2γ )F α γ ) F γf. 2.4 Since F 2 ω n n / nμ μ nν ν, so, H 2 ) ω n n. 2.4 Therefore, for, H 2 ) ω n n 2 ) 2 ) δ This shows that the result is sharp. 9. Letting γ intheorem 2.3 above, we obtain the following result of Kim and Rønning

10 Abstract and Applied Analysis Corollary 2.4. Let δ< and α, and define β β δ by β δ / t dt. t 2.43 If f W β α, P α β,thenv λ f W δ α, P α δ. The value of β is sharp. Upon setting c t c with <c, we have the following corollary. Corollary 2.5. Let δ<, α, γ, and <c be given, and let G be defined by G c c u c f u du Suppose that f W β α, γ, theng W δ α,, where β 2 c 2F, 2 c;3 c, 2 c c 2 F, 2 c;3 c, The constant β is sharp.. The special case of Corollary 2.5 with γ has been obtained by Aghalary et al. References R. Fournier and S. Ruscheweyh, On two extremal problems related to univalent functions, The Rocky Mountain Mathematics, vol. 24, no. 2, pp , R. W. Barnard, S. Naik, and S. Ponnusamy, Univalency of weighted integral transforms of certain functions, Computational and Applied Mathematics, vol. 93, no. 2, pp , S. Ponnusamy and F. Rønning, Integral transforms of a class of analytic functions, Complex Variables and Elliptic Equations, vol. 53, no. 5, pp , R. M. Ali, A. O. Badghaish, V. Ravichandran, and A. Swaminathan, Starlikeness of integral transforms and duality, Mathematical Analysis and Applications, vol. 385, no. 2, pp , S. Ponnusamy, Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane, The Rocky Mountain Mathematics, vol. 28, no. 2, pp , 998, Reports of the Department of Mathematics, Preprint 92, University of Helsinki, Finland, S. Ruscheweyh, Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc, Transactions of the American Mathematical Society, vol. 2, pp , S. E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Transactions of the American Mathematical Society, vol. 38, no. 2, pp. 3 34, S. Ruscheweyh, Convolutions in Geometric Function Theory, vol. 83 of Séminaire de Mathématiques Supérieures, Presses de l UniversitédeMontréal, Montreal, Canada, Y. C. Kim and F. Rønning, Integral transforms of certain subclasses of analytic functions, Mathematical Analysis and Applications, vol. 258, no. 2, pp , 2. K. Noshiro, On the theory of schlicht functions, the faculty of science, Hokkaido University, vol. 2, pp , 934. R. Aghalary, A. Ebadian, and S. Shams, Geometric properties of some linear operators defined by convolution, Tamkang Mathematics, vol. 39, no. 4, pp , 28.

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