Research Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions

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1 Abstract and Applied Analysis Volume 20, Article ID 59405, 0 pages doi:0.55/20/59405 Research Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions Firas Ghanim and Maslina Darus 2 Faculty of Management, Multimedia University, Cyberjaya, 6300 Selangor, Malaysia 2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia Correspondence should be addressed to Maslina Darus, maslina@ukm.my Received 27 March 20; Accepted 7 July 20 Academic Editor: Yoshikazu Giga Copyright q 20 F. Ghanim 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 introduce a new subclass of meromorphically analytic functions, which is defined by means of a Hadamard product or convolution. A characterization property such as the coefficient bound is obtained for this class. The other related properties, which are investigated in this paper, include the distortion and the radius of starlikeness. We also consider several applications of our main results to the generalized hypergeometric functions.. Introduction Let A be the class of functions f which are analytic in the open unit disk U {z C : z < }.. As usual, we denote by S the subclass of A, consisting of functions which are also univalent in U. Let w be a fixed point in U and A w {f H D : f w f w 0}. In, Kanas and Ronning introduced the following classes

2 2 Abstract and Applied Analysis S w { f A w : f is univalent in U }, { ( z w f ) } ST w f A w :Re > 0, z U, { ( f CV w f A : Re z w f ) } f > 0, z U..2 Later, Acu and Owa 2 studied the classes extensively. The class ST w is defined by geometric property that the image of any circular arc centered at w is starlike with respect to f w, and the corresponding class S c w is defined by the property that the image of any circular arc centered at w is convex. We observed that the definitions are somewhat similar to the ones introduced by Goodman in 3, 4 for uniformly starlike and convex functions except that, in this case, the point w is fixed. Let Σ w denote the subclass of A w consisting of the function of the form f z w a n z w n. n.3 The functions f in Σ w are said to be starlike functions of order β if and only if { Re z w f } >β z w U,.4 f for some β 0 β <. We denote by S w β the class of all starlike functions of order β. Similarly, a function f in S w is said to be convex of order β if and only if ( Re z w f ) f >β z w U,.5 for some β 0 β<. We denote by C w β the class of all convex functions of order β. For the function f Σ w, we define I 0 λf f, I λ f z w f 2 z w, ) I 2 λ (I f z w 2 f z w,.6

3 Abstract and Applied Analysis 3 and, for k, 2, 3,..., we can write ) I k λ (I f z w k 2 f z w z w λ n k a n z w n, n.7 where λ, k 0and z w U. The differential operator I k is studied extensively by Ghanim and Darus 5, 6 and Ghanim et al. 7. The Hadamard product or convolution of the functions f given by.3 with the function g and h given, respectively, by g z w b n z w n, n h z w c n z w n, n.8 can be expressed as follows: ( ) f g z w a n b n z w n, n ( ) f h z w a n c n z w n. n.9 Suppose that f and g are two analytic functions in the unit disk U. Then, we say that the function g is subordinate to the function f, and we write g f z U,.0 if there exists a Schwarz function ϖ with ϖ 0 0and ϖ < such that g f ϖ z U.. By applying the above subordination definition, we introduce here a new class Σ w A, B, k, α, λ of meromorphically functions, which is defined as follows: Definition.. A function f Σ w of the form.3 is said to be in the class Σ w A, B, k, α, λ if it satisfies the following subordination property: α Ik λ I k λ ( ) f g ( ) α f h z w B z w z w U,.2 where B<A, k 0, α > 0, λ, with condition I k λ f h / 0.

4 4 Abstract and Applied Analysis The purpose of this paper is to investigate the coefficient estimates, distortion properties, and the radius of starlikeness for the class Σ w A, B, k, α, λ. Some applications of the main results involving generalized hypergeometric functions are also considered. 2. Characterization and Other Related Properties In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function f Σ w of the form.3 to belong to the class Σ w A, B, k, α, λ of meromorphically analytic functions. Theorem 2.. The function f Σ w is said to be a member of the class Σ w A, B, k, α, λ if it satisfies λ n k αb n B c n α B A B a n A B. n 2. The equality is attained for the function f n given by f n z w λ n k αb n B c n α B A B z w n. 2.2 Proof. Let f Σ w A, B, k, α, λ, and suppose that α Ik λ I k λ ( ) f g ( ) α f h z w B z w. 2.3 Then, in view of 2.2, we have α n λ n k a n b n c n z w n n λ n k a n αbb n { αb}c n z w n α n λ n k a n b n c n z w n n λ n k a n αbb n { αb}c n z w n. 2.4 Letting z w, we get λ n k αb n B c n α B A B a n, n 2.5 which is equivalent to our condition of the theorem, so that f Σ w A, B, k, α, λ. Hence we have the theorem.

5 Abstract and Applied Analysis 5 Theorem 2. immediately yields the following result. Corollary 2.2. If the function f Σ w belongs to the class Σ w A, B, k, α, λ, then a n λ n k αb n B c n α B A B, 2.6 n, where the equality holds true for the functions f n given by 2.2. We now state the following growth and distortion properties for the class Σ w A, B, k, α, λ. Theorem 2.3. If the function f defined by.3 is in the class Σ w A, B, k, α, λ, then for 0 < z w r<, one has r αb B c α B A B r f r αb B c α B A B r, r 2 αb B c α B A B 2.7 f r 2 αb B c α B A B. Proof. Since f Σ w A, B, k, α, λ, Theorem 2. readily yields the inequality a n αb B c α B A B. n 2.8 Thus, for 0 < z w r< and utilizing 2.8, we have

6 6 Abstract and Applied Analysis f f m z w a n z w n r r m r a n n n αb B c α B A B r, m z w a n z w n r r m r a n n n αb B c α B A B r. 2.9 Also, from Theorem 2.,weget na n αb B c α B A B. n 2.0 Hence f z w 2 m r na n n m na n z w n n r 2 f z w 2 αb B c α B A B, m na n z w n n 2. m r na 2 n r 2 n αb B c α B A B. This completes the proof of Theorem 2.3.

7 Abstract and Applied Analysis 7 We next determine the radius of meromorphically starlikeness of the class Σ w A, B, k, α, λ, which is given by Theorem 2.4. Theorem 2.4. If the function f defined by.3 is in the class Σ w A, B, k, α, λ, then f is meromorphically starlike of order δ in the disk z w <r,where { } δ αbn B c n α B A B / n r inf. 2.2 n n 2 δ The equality is attained for the function f n given by 2.2. Proof. It suffices to prove that z w ( I k f ) I k f δ. 2.3 For z w <r, we have z w ( I k f ) I k f n n λ n k a n z w n / z w n λ n k a n z w n n n λ n k a n z w n n λ n k a n z w n n n λ n k a n z w n n λ n. k a n z w n 2.4 Hence 2.4 holds true for ( ) n λ n k a n z w n δ λ n k a n z w n n n 2.5 or n n 2 δ λ n k a n z w n δ. 2.6 With the aid of 2. and 2.6, itistruetosaythatforfixedn n 2 δ λ n k z w n δ λ n k αb n B c n α B A B n. 2.7

8 8 Abstract and Applied Analysis Solving 2.7 for z w,weobtain { } δ αbn B c n α B A B / n z w <. 2.8 n 2 δ This completes the proof of Theorem Applications Involving Generalized Hypergeometric Functions Let us define the function φ a, c; z by φ a, c; z z w n 0 a n c n a n z w n, 3. for c / 0,, 2,...,anda C/{0}, where λ n λ λ n is the Pochhammer symbol. We note that φ a, c; z z w 2 F,a,c; z, 3.2 where 2F b, a, c; z n 0 b n a n c n z w n. 3.3 n! Corresponding to the function φ a, c; z and using the Hadamard product which was defined earlier in the introduction section for f Σ, we define here a new linear operator L a, c on Σ by L w a, c f φ a, c; z f z w n a n c n a n z w n. 3.4 For a function f L w a, c f, we define I 0( L w a, c f ) L w a, c f, 3.5 and, for k, 2, 3,..., I k( L w a, c f ) ( ) z I k L 2 a, c f z w z w n k a n a n z w n. n c n 3.6

9 Abstract and Applied Analysis 9 We note I k L w a, a f studied by Ghanim and Darus 5, 6 and Ghanim et al. 7, and also, I k L 0 a, c f studied by Ghanim and Darus 8, 9 and Ghanim et al. 0. The subordination relation.2 in conjunction with 3.4 and 3.6 takes the following form: α Ik L w a,c f I k L w a, c f α z w B z w B<A, k 0, α > 0. Definition 3.. A function f Σ w of the form.3 is said to be in the class Σ w A, B, k, α, a, c if it satisfies the subordination relation 3.7 above. Theorem 3.2. The function f Σ w is said to be a member of the class Σ w A, B, k, α, a, c if it satisfies n k αb n B c n α B A B a n c n a n. n 3.8 The equality is attained for the function f n given by f n z w c n n k αb n B c n α B A B a n z w n, n 3.9 n. Proof. By using the same technique employed in the proof of Theorem 2. along with Definition 3., we can prove Theorem 3.2. The following consequences of Theorem 3.2 can be deduced by applying 3.8 and 3.9 along with Definition 3.. Corollary 3.3. If the function f Σ w belongs to the class Σ w A, B, k, α, a, c, then a n c n n k αb n B c n α B A B a n, 3.0 n, where the equality holds true for the functions f n given by 3.9. Corollary 3.4. If the function f defined by.3 is in the class Σ w A, B, k, α, a, c, thenf is meromorphically starlike of order δ in the disk z w <r 3,where { } δ αbn B c n α B A B c r 3 inf n / n. 3. n n 2 δ a n The equality is attained for the function f n given by 3.9.

10 0 Abstract and Applied Analysis A slight background related to the formation of the present operator can be found in, and other work can be tackled using this type of operator. Also, the meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava 2, 3,Liu 4, Liu and Srivastava 5, and Cho and Kim 6. Acknowledgment The work presented here was fully supported by UKM-ST-06-FRGS References S. Kanas and F. Ronning, Uniformly starlike and convex functions and other related classes of univalent functions, Annales Universitatis Mariae Curie-Sklodowska, vol. 53, pp , M. Acu and S. Owa, On some subclasses of univalent functions, Inequalities in Pure and Applied Mathematics, vol. 6, no. 3, pp. 6, A. W. Goodman, On uniformly starlike functions, Mathematical Analysis and Applications, vol. 55, no. 2, pp , A. W. Goodman, On uniformly convex functions, Annales Polonici Mathematici, vol.56,no.,pp , F. Ghanim and M. Darus, On certain class of analytic function with fixed second positive coefficient, International Mathematical Analysis, vol. 2, no. 2, pp , F. Ghanim and M. Darus, Some subordination results associated with certain subclass of analytic meromorphic functions, Mathematics and Statistics, vol. 4, no. 2, pp. 2 6, F. Ghanim, M. Darus, and S. Sivasubramanian, On new subclass of analytic univalent function, International Pure and Applied Mathematics, vol. 40, no. 3, pp , F. Ghanim and M. Darus, Linear operators associated with a subclass of hypergeometric meromorphic uniformly convex functions, Acta Universitatis Apulensis, no. 7, pp , F. Ghanim and M. Darus, Certain subclasses of meromorphic functions related to Cho-Kwon- Srivastava operator, Far East Mathematical Sciences, vol. 48, no. 2, pp , F. Ghanim, M. Darus, and A. Swaminathan, New subclass of hypergeometric meromorphic functions, Far East Mathematical Sciences, vol. 34, no. 2, pp , B. A. Frasin and M. Darus, On certain meromorphic functions with positive coefficients, Southeast Asian Bulletin of Mathematics, vol. 28, no. 4, pp , J. Dziok and H. M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Advanced Studies in Contemporary Mathematics, vol. 5, no. 2, pp. 5 25, J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms and Special Functions, vol. 4, no., pp. 7 8, J. L. Liu, A linear operator and its applications on meromorphic p-valent functions, Bulletin of the Institute of Mathematics, vol. 3, no., pp , J. L. Liu and H. M. Srivastava, Certain properties of the Dziok-Srivastava operator, Applied Mathematics and Computation, vol. 59, no. 2, pp , N. E. Cho and I. H. Kim, Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function, Applied Mathematics and Computation, vol. 87, no., pp. 5 2, 2007.

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