ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS

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1 TWMS J. App. Eng. Math. V.4, No.1, 214, pp ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS F. GHANIM 1 Abstract. This paper illustrates how some inclusion relationships of certain class of univalent meromorphic functions may be defined by using the linear operator. Further, a property preserving integrals is considered for the final outcome of the study. Keywords:Analytic Function; Meromorphic Function; Integral Operator; Linear Operator; Hadamard Product; Hypergeometric Function. AMS Subject Classification: 3C45, 3C5. 1. Introduction A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function f) is a function of the form f ) = g ) h ), where g) and h) are entire functions with h) see [1], p. 64). A meromorphic function therefore may only have finite-order, isolated poles and eros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by csc 1 on the punctured disk U = { : < < 1}. An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example the Gamma function is meromorphic in the whole complex plane, see [9] and [1]. In this paper, the linear operator is used to define some inclusion relationships of mermorophic fucntions. Moreover, the final outcome of the study has a property preserving integrals considered. 1 University of Sharjah, College of Sciences, Department of Mathematics, Sharjah, United Arab Emirates, fgahmed@sharjah.ac.ae Submitted for GFTA 13, held in Işık University on October 12, 213. TWMS Journal of Applied and Engineering Mathematics, Vol.4, No.1; c Işık University, Department of Mathematics 214; all rights reserved. 45

2 46 TWMS J. APP. ENG. MATH. V.4, NO.1, Preliminaries Let Σ denote the class of meromorphic functions f) normalied by f) = 1 + a n n, 1) which are analytic in the punctured unit disk U = { : < < 1}. For β, we denote by S β) and kβ), the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β and convex of order β in U. For functions f j )j = 1; 2) defined by f j ) = 1 + a n,j n, 2) we denote the Hadamard product or convolution) of f 1 ) and f 2 ) by f 1 f 2 ) = 1 + a n,1 a n,2 n. 3) Analogous to the integral operator defined by Jung et al. [3] introduced and investigated the following integral operator: Q : Σ Σ defined, in terms of the familiar Gamma function, by = 1 By setting f ) = 1 + Γ β + α) Q f ) = Γ α) Γ β) + Γ β + α) Γ β) Γ β) Γ β + α) 1 β+1 t β 1 t ) α 1 f t) dt Γ n + β + 1) Γ n + β + α + 1) k, α > ; β > ; U ). 4) Γ n + β + α + 1) k, α > ; β > ; U ), 5) Γ n + β + 1) we define a new function f λ ) in terms of the Hadamard product or convolution): f ) f λ ) = 1 1 ) λ, α > ; β > ; λ > ; U ). 6) Then, motivated essentially by the operator Q, we now introduce the operator which is defined as Let us put Q λ : Σ Σ Q λ := f λ ) f ), α > ; β > ; λ > ; U, f Σ). 7) q λ, µ ) = 1 + ) λ µ n, λ >, µ ). 8) n λ

3 F. GHANIM: ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC Corresponding to the functions Q λ and q λ, µ ), and using the Hadamard product again for f) Σ, we define a new linear operator f) = 1 Γ β + α) ) a) µ + n+1 Γ n + β + 1) λ a n n 9) Γ β) n + 1)! Γ n + β + α + 1) n λ U ), where is a) n the Pochhammer symbol defined by { 1, n = a) n := a a + 1)... a + n 1) n := {1, 2,...}). Clearly, Q 1,λ, = Q. The meromorphic functions with the integral operators were considered recently by [1],[2], [5], [6] and [7]. It is readily verified from 9) that f ) ) = aq a+1,λ,µ f) a + 1) f), 1) Q λ,µ α+1,β f ) = β + α) Q a,λ,µ f) β + α + 1) Qa,λ,µ f). 11) Definition 2.1. We say that a function f Σ is in the class Σ a,λ,µ γ) if it satisfies the following condition: { ) } R 2 f) > γ, U ) 12) where α >, β >, λ >, µ, and γ < 1. Using 1) condition 12) can be written in the form { R aq a+1,λ,µ } f) + a + 1) f) > γ γ < 1, U. 13) 3. Main results We will assume in the reminder of this paper that Σ a,λ,µ γ). We begin by recalling the following result Jack s lemma), which we shall apply in proving our inclusion theorems below. Lemma 3.1. [4] Let the nonconstant) function w) be analytic in U, with w) =. If w ) attains its maximum value on the circle = r < 1 at a point U, then w ) = ξw ), where ξ is a real number and ξ 1. Theorem 3.1. The following inclusion property holds true for the class Σ a,λ,µ γ) Σ a+1,λ,µ γ) Σ a,λ,µ γ). 14) Proof. Let f) Σ a+1,λ,µ γ) and define a regular function w) in U such that w) =, w) 1 by aq a+1,λ,µ f) + a + 1) 1 + 2γ 1) w) f) =. 15) 1 + w)) Differentiating 15) with respect to, we obtain ) 2 Q a+1,λ,µ 1 + 2γ 1) w) f) = 1 + w) 2 1 γ) λ w ) 1 + w )) 2. 16)

4 48 TWMS J. APP. ENG. MATH. V.4, NO.1, 214 We claim that w) < 1 for U. Otherwise there exists a point U such that max w ) = w) = 1. Applying Jack s lemma, we have w ) = ξw ), ξ 1. 17) From 16) and 17) we have ) 2 Q a+1,λ,µ 1 + 2γ 1) w ) f) = 1 + w ) { } Since R 1+2γ 1)w ) 1+w ) = γ, ξ 1 and w ) 1+w )) 2 { R 2 Q a+1,λ,µ 2 1 γ) λ w ) 1 + w )) 2. 18) is real and positive, we see that ) } f) < γ, which obviously contradicts f) Σ a+1,λ,µ γ). Hence w) < 1 for U, and it follows from 15) that f) Σ a,λ,µ γ). This completes the proof of Theorem 3.1. Theorem 3.2. Let c be any real number and c >. If f) Σ a,λ,µ γ), then J c ) = c c t c ft)dt Σ a,λ,µ γ), c > ). 19) Proof. From 19), we have J c) = cq a+1,λ,µ J c ) c + 1) J c). 2) Define a regular function w) in U such that w) =, w) 1 by 2 From 2) and 21) we have cq a+1,λ,µ ) J) 1 + 2γ 1) w) =. 21) 1 + w) J) c + 1) 1 + 2γ 1) w) J) =. 22) 1 + w)) Differentiating 22) with respect to, and using 21) we obtain 2 J) 1 + 2γ 1) w) 2 1 γ) w ) = 1 + w) c 1 + w )) 2. 23) The remaining part of the proof of Theorem 3.2 is similar to that of Theorem 3.1. Theorem 3.3. If f) Σ a+1,λ,µ γ), and satisfy the condition { ) } R 2 Q a+1,λ,µ 1 γ) f) > γ c > ). 24) 2c Then the function J c ) = c c t c ft)dt Σ a,λ,µ γ) c > ). Proof. The proof of Theorem 3.3 is similar to that of Theorem 3.2 and hence, it will not be elaborated.

5 F. GHANIM: ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC Theorem 3.4. Let f) be defined by J c ) = c c t c ft)dt Σ a,λ,µ γ) c > ). 25) If J c ) Σ a,λ,µ γ), then f ) Σa,λ,µ γ) in < c 1+ c Proof. Since J c ) Σ a,λ,µ γ) we can write J) γ + 1 γ) u ) =, 26) where u ) P, the class of functions with positive real part in the unit disk U and normalied by u) = 1. We can re-write 26) as aq a+1,λ,µ J) + a + 1) γ + 1 γ) u ) J) = 27) Differentiating 27) with respect to, and using 2) we obtain 2 J) γ = u ) γ) c u ). 28) Using the well-known estimate see[[11]]) u ) 2 J) γ R 1 γ) 1 2r 1 r 2 R u ), = r 28) yields ) 2r c 1 r 2 R u ) 29) ) The right-hand side of 29) is positive if r < 1+. This completes the proof of Theorem c c References [1] Lashin, A. Y., 1993), On certain subclasses of meromorphic functions associated with certain integral operators, Comput. Math. Appl., 1761), [2] Ghanim, F. and Darus, M., 211), A new class of meromorphically analytic functionswith applications to generalied hypergeometric functions, Abstract and Applied Analysis, 211, Article ID 15945, 1 pages, doi:1.1155/211/ [3] Jung, I. B., Kim, Y. C. and Srivastava, H. M., 23),The Hardy space of analytic functions associated with certain one-parameter families of integral operators, Journal ofmathematical Analysis and Applications, 24, [4] Jack, I. S., 1971), Functions starlike and convex of order, J. London Math. Soc., 32), [5] Liu, J. L., 21),The Noor integral operator and strongly starlike functions, J. Math. Anal. Appl., 261, [6] Noor, K. I., On new classes of integral operators, J. Natur. Geom., ), [7] Noor, K. I., and Noor, M. A., 1999), On integral operators, J. Natur. Geom., 238, [8] K. Knopp, Meromorphic functions., Ch. 2 in theory of functions parts I and II, two volumes bound as one, part II., New York: Dover 1996),pp [9] Kumar Pandey, R., 28), Applied Complex Analysis, Discovery Publishing House, Grand Rapids, Michigan. [1] Krant, S. G., 1999), Meromorphic functions and singularities at infinity, Handbook of Complex Variables., Boston, MA: Birkhuser,pp [11] Nehari, Z.,1952), Conformal Mapping, McGraw-Hill Book Company, New York, Toronto and London. [12] Wang, Z. G., Liu, Z.H. and Sun, Y., 29),Some subclasses of meromorphic functions associated with a family of integral operators, J. Inequal. Appl., Article ID 93123, 1-18.