SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION

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1 Volume 29), Issue, Article 4, 7 pp. SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION R. K. RAINA / GANPATI VIHAR, OPPOSITE SECTOR 5 UDAIPUR 332, RAJASTHAN, INDIA rkraina_7@hotmail.com Receive 4 August, 28; accepte January, 29 Communicate by H.M. Srivastava ABSTRACT. A class of fractional erivative operators with the Appell hypergeometric function in the kernel) is use here to efine a new subclass of analytic functions an a coefficient boun inequality is establishe for this class of functions. Also, an inclusion theorem for a class of fractional integral operators involving the Hary space of analytic functions is prove. The concluing remarks briefly mentions the relevances of the main results an possibilities of further work by using these new classes of fractional calculus operators. Key wors an phrases: Analytic functions, Hary space, Fractional erivatives an fractional integrals, Appell hypergeometric function, Inclusion relation. 2 Mathematics Subject Classification. 26A33, 3C45.. INTRODUCTION, DEFINITIONS AND PRELIMINARIES Let An) enote the class of functions f) normalie by.) f) = + a k k n N), which are analytic in the open unit isk U = : C an <. We enote by α,α,β,β,γ) n σ) the subclass of functions in An) which also satisfy the inequality:.2) Re χ α, α, β, β, γ) α+α +γ D α,α,β,β,γ), f) > σ U), 23-8

2 2 R. K. RAINA where D α,α,β,β,γ), is the generalie fractional erivative operator efine below), an for convenience).3) χ m α, α, β, β, γ) provie that.4) = Γ + m + β )Γ + m α α γ)γ + m α β γ) Γ + m)γ + m α + β )Γ + m α α β γ) σ < ; γ < ; γ< min α α, α β, α α β) + m + ; β > max, α ) m. m N), Following [8], a function f) is sai to be in the class V n θ k ) if f) An) satisfies the conition that arga k ) = θ k k n + ; n N) an if there exists a real number ρ such that.5) θ k + k )ρ πmo2π) k n + ; n N), then we say that f) is in the class V n θ k ; ρ). Suppose V n = V n θ k ; ρ) over all possible sequences θ k with ρ satisfying.5), then we enote by α,α,β,β,γ) n σ) the subclass of V n which consists of functions f) belonging to the class α,α,β,β,γ) n σ). We present here the following family of fractional integral an erivative) operators which involve the familiar Appell hypergeometric function F 3 see also Kiryakova [4] an Saigo an Maea [9]). Definition.. Let γ > an α, α, β, β R. Then the fractional integral operator I α,α,β,β,γ), of a function f) is efine by.6) I α,α,β,β,γ), f) = α Γγ) ζ) γ ζ α F 3 α, α, β, β ; γ; ζ, ) fζ)ζ γ> ), ζ where the function f) is analytic in a simply-connecte region of the complex -plane containing the origin, an it is unerstoo that ζ) γ enotes the principal value for arg t) < 2π. The function F 3 occurring in the kernel of.6) is the familiar Appell hypergeometric function of thir type also known as Horn s F 3 - function; see, for example, []) efine by.7) F 3 α, α, β, β ; γ;, ξ) = m= n= α) m α ) n β) m β ) n γ) m+n m ξ n m! n! <, ξ < ), which is relate to the Gaussian hypergeometric function 2 F α, β; γ; ) by the following relationship: 2F α, β; γ; ) = F 3 α, α, β, β ; γ;, ) = F 3 α,, β, β ; γ;, ξ) = F 3 α, α, β, ; γ;, ξ). Definition.2. The fractional erivative operator D α,α,β,β,γ), of a function f) is efine by.8) D α,α,β,β,γ), f) = n n Iα,α,β n,β,n γ), f) n γ < n; n N). J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.

3 SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION 3 It may be observe that for.9) α = λ + µ, α = β =, β = η, γ = λ, we obtain the relationship.) I λ+µ,, η,,λ), = I λ,µ,η, in terms of the Saigo type fractional integral operator I λ,µ,η, [2]). On the other han, if.) α = µ λ, α = β =, β = η, γ = λ, then we get.2) D µ λ,, η,,λ), = J λ,µ,η,, where J λ,µ,η, is the Saigo type fractional erivative operator [6]; see also [7]). Further, when.3) α = β =, α = µ, γ = λ or λ), then the operators I, µ,,,λ), an D, µ,,, λ), correspon to the ifferential-integral operators Q λ µ ue to Diok [2]. Let H p p < ) be the class of analytic functions in U such that.4) f p = lim r M pr, f) <, where.5) f p = 2π 2π sup f ). r fre iθ ) p) p < p < ), In this paper we first efine a new function class in terms of the fractional erivative operators with the Appell hypergeometric function in the kernel) an then establish a coefficient boun inequality for this function class. Also, we prove an inclusion theorem for a class of fractional integral operators involving the Hary space of analytic functions. The relevance of the main results an possibilities of further work by using the new classes of fractional calculus operators are briefly pointe out in the concluing section of this paper. 2. A SET OF COEFFICIENT BOUNDS We begin by proving the following coefficient bouns inequality for a function f) to be in the class α,α,β,β,γ) n σ). Theorem 2.. Let f) efine by.) be in the class α,α,β,β,γ) n 2.) a k χ k α, α, β, β, γ) σ χ α, α, β, β, γ), where χ m α, α, β, β, γ) is efine by.3). The result is sharp. σ), then Proof. Assume that Re χ α, α, β, β, γ) α+α γ D α,α,β,β,γ), f) > σ U). J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.

4 4 R. K. RAINA Using.) an the formula see, e.g. [9, p. 394]): 2.2) D α,α,β,β,γ), q we obtain = 2.3) Re Γ + q)γ + q α + β )Γ + q α β γ) Γ + q + β )Γ + q α β γ)γ + q α α γ) q α α γ, γ < ; α, α, β, β R; q > max, α β, α + β + γ) ) + χ α, α, β, β, γ) χ k α, α, β, β, γ) a k k > σ U), an for f) V n θ k ; ρ) = re iθ ), the inequality thus obtainable from 2.3) on letting r therein, reaily yiels χ α, α, β, β, γ) 2.4) Re + χ k α, α, β, β, γ) a k exp i θ k + k ) ρ)) > σ. If we apply.5), then 2.4) gives 2.5) χ α, α, β, β, γ) χ k α, α, β, β, γ) a k > σ, which leas to the esire inequality 2.). We also observe that the equality sign in 2.) is attaine for the function f) efine by 2.6) f) = + σ)χ kα, α, β, β, γ) k exp iθ χ α, α, β, β k ), γ) an this completes the proof of Theorem 2.. k n + ; n N), 3.) Uner the hypotheses of Definition., let 3. INCLUSION RELATIONS γ > ; min γ α α, γ α β, β, γ α α β, β α ) > 2; then the fractional integral operator is efine by 3.2) Ω α,α,β,β,γ) α, α, β, β R, Ω α,α,β,β,γ) : A A A) = A) f) = χ α, α, β, β, γ) α+α +γ I α,α,β,β,γ),) f). where χ α, α, β, β, γ) is given by.3). By using the formula [9, p. 394]; see also [4, p. 7, Lemma 9]) 3.3) I α,α,β,β,γ), q = Γ + q)γ + q α + β )Γ + q α α β + γ) Γ + q + β )Γ + q α β + γ)γ + q α α + γ) q α α +γ, γ > ; α, α, β, β R; q > max, α β, α + β γ) ) J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.

5 SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION 5 it follows from.), 3.2) an 3.3) that 3.4) Ω α,α,β,β,γ) f) = + χ α, α, β, β, γ) k=2 a k χ k α, α, β, β, γ) k, where as before) χ k α, α, β, β, γ) is given by.3). Before stating an proving our main inclusion theorem, we recall here the following known results concerning the class Rρ) in A which satisfies the inequality that Rf ) > ρ ρ < ), where R) is enote by R. Lemma 3. [3, p. 4]). Let f) R, then 3.5) f) H p : < p < ). Lemma 3.2 [5, p. 533]). Let f) efine by.) be in the class Rρ) ρ < ), then 3.6) a k 2 k k = 2, 3, 4,...). Theorem 3.3. Let f) R, then uner the constraints state in 3.)) 3.7) Ω α,α,β,β,γ) f) H p < p < ) an 3.8) Ω α,α,β,β,γ) f) H γ > ). Proof. In view of.6) an 3.2), we obtain 3.9) Ω α,α,β,β,γ) This implies that 3.) Re f) = χ α, α, β, β, γ) Ωα,α,β,β,γ) f) = χ α, α, β, β, γ) Since f) R, therefore, we infer from 3.) that t) γ t α F 3 α, α, β, β ; γ; t, t ) ft)t. t) γ t α F 3 α, α, β, β ; γ; t, ) R f t) t. t 3.) Ω α,α,β,β,γ) f) R, an applying Lemma 3., 3.) gives the inclusion relation 3.7) uner the conitions state in 3.). To prove the result 3.8), we observe the following three-term recurrence relation: 3.2) Ωα,α,β,β,γ) f) = γ α β + ) Ω α,α,β,β,γ ) f) γ α β) Ω α,α,β,β,γ) f), which yiels the inequality p 3.3) Ωα,α,β,β,γ) f) r p γ α β + ) p Ω α,α,β,β,γ ) f) p γ α β) p Ω α,α,β,β,γ) f) p = r), J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.

6 6 R. K. RAINA provie that 3.4) γ > ; min +γ α α, +γ α β, +β, +γ α α β, +β α ) > ; α, α, β, β R an < p <. Making use of.4) an.5), the above inequality 3.3) with p = ) yiels 3.5) M r, ) ),β,β,γ) Ωα,α f) r γ α β + ) M r, Ω α,α,β,β,γ ) f) ) γ α β) M r, Ω α,α,β,β,γ) f) an 3.6) Ωα,α,β,β,γ) f) γ α β + ) f) γ α β) Ω α,α,β,β,γ ) Applying 3.7), we infer uner the constraints state in 3.4)) that Ω α,α,β,β,γ) 3.7) Ω α,α,β,β,γ ) f) H an Ω α,α,β,β,γ) f) H γ > ), an consequently 3.6) implies that f) H, Ωα,α,β,β,γ) f). provie that the conitions state in 3.4) are satisfie. By appealing to a known result [, p. 42, Theorem 3.], we infer from 3.7) that Ω α,α,β,β,γ) f) is continuous in U = : C an. But U being compact, we finally conclue that Ω α,α,β,β,γ) f) is a boune analytic function in U, an the proof of the secon assertion 3.8) of Theorem 3.3 is complete. The assertion 3.8) of Theorem 3.3 can also be prove by applying Lemma 3.2 see also [3, p. 45]). Inee, it follows from 3.4) an 3.6) that Ω α,α,β,β,γ) f) + χ α, α, β, β, γ) + 2 : χ α, α, β, β, γ) = + k=2 a k k χ k α, α, β, β, γ) k=2 Γk) Γk + )χ k α, α, β, β, γ) 22 α + β )2 α α β + γ) 2 + β )2 α α + γ)2 α β + γ) 4 F 3 [, 2, 3 α + β, 3 α α β + γ; 3 + β, 3 α α + γ, 3 α β + γ; in terms of the generalie hypergeometric function. Now, for fixe values of the parameters α, α, β, β, γ satisfying the conitions state in 3.), we observe that by using the asymptotic formula [, p. 9], Γk) Γk + )χ k α, α, β, β, γ) = o k γ ) an since γ >, this proves our assertion 3.8). k ), ] J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.

7 SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION 7 4. CONCLUDING REMARKS In view of the relationships.) an.2), the main results Theorems 2. an 3.3) of this paper woul correspon to the results ue to Raina an Srivastava [8, p. 75, Theorem ; p. 79, Theorem 7]. Furthermore, in view of the relationship.3), we can easily apply Theorems 2. an 3.3 to obtain the corresponing results associate with Diok s ifferential-integral operators [2]. The family of fractional calculus operators fractional integrals an fractional erivatives) efine by.6) an.8) can fruitfully be use in Geometric Function Theory. Several new analytic, multivalent or meromorphic) function classes can be efine an the various properties of coefficient estimates, istortion bouns, raii of starlikeness, convexity an close to convexity for such contemplate classes investigate. REFERENCES [] P.L. DUREN, Theory of H p Spaces, Vol. 38, A series of monographs an textbooks in pure an applie mathematics, Acaemic Press, New York, 97. [2] J. DZIOK, Applications of the Jack lemma, Acta Math. Hungar., 5 24), [3] I. B. JUNG, Y. C. KIM AND H. M. SRIVASTAVA, The Hary space of analytic functions associate with certain one-parameter families of integral operators, J. Math. Anal. Appl., ), [4] V. KIRYAKOVA, On two Saigo s fractional integal operators in the class of univalent functions, Fracl. Cal. Appl. Math., 9 26), [5] T.H. MACGREGOR, Functions whose erivative has a positive real part, Trans. Amer. Math. Soc., 4 962), [6] R.K. RAINA AND T.S. NAHAR, Characteriation properties for starlikeness an convexity of some subclasses of analytic functions involving a class of fractional erivative operators, Acta Math. Univ. Comenianae, 69 2), 8. [7] R.K. RAINA AND H.M. SRIVASTAVA, A certain subclass of analytic functions associate with operators of fractional calculus, Comput. Math. Appl., ), 3 9. [8] R.K. RAINA AND H.M. SRIVASTAVA, Some subclasses of analytic functions associate with fractional calculus operators, Comput. Math. Appl., ), [9] M. SAIGO AND N. MAEDA, More generaliation of fractional calculus, In:Transform Methos an Special Functions, Varna 96 Proc. Secon Internat. Workshop), Science Culture Technology Publishing, Singapore 998), [] H.M. SRIVASTAVA AND P.W. KARLSSON, Multiple Gaussian Hypergeometric Series, Halste Press Ellis Horwoo Limite, Chichester), John Wiley an Sons, New York, 985. [] H.M. SRIVASTAVA AND S. OWA, Current Topics in Analytic Function Theory, Worl Scientific Publishing Company, Singapore, New Jersey, Lonon an Hongkong, 992. [2] H.M. SRIVASTAVA, M. SAIGO AND S. OWA, A class of istortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl., 3 988), J. Inequal. Pure an Appl. Math., ) 29), Art. 4, 7 pp.