NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS

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1 Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS M. ALBEHBAH 1 AND M. DARUS 2 Abstract. In this aer, we introduce a new class (A, B, k) a,c for 1 B < A 1 which consists of hyergeometric meromorhic functions of the form L (a, c)f(z) = 1 z + a n+ z n+ in U = {z : 0 < z < 1}. We determine sufficient conditions, distortion roerties, radii of starlikeness and convexity and inclusion roerties for the class L (a, c)f(z). Let 1. Introduction denote the class of meromorhic functions f normalized by f(z) = 1 z + a n+ z n+, which are analytic and -valent in the unctured unit disk U = {z : 0 < z < 1}. For 0 α <, we denote by S (α) and K (α), the subclasses of consisting of all meromorhic functions which are, resectively, starlike of order α and convex of order α in U. The classes S (α), K (α) and various other subclasses of have been studied rather extensively by Aouf et al. [3 5], Ghanim and Darus [16], Srivastava [19], Kulkarni et al. [20], Morga [26, 27], Owa et al. [28], Srivastava and Owa [29], Uralegaddi and Somantha [31, 32], and Yang [33]. Key words and hrases. -valent hyergeometric functions, meromorhic functions, starlike functions, convex functions, Hadamard roduct Mathematics Subject Classification. Primary: 30C45. Secondary: 33C20, 30C85. Received: January 17, Acceted: October 12,

2 84 M. ALBEHBAH AND M. DARUS For functions f j, (j = 1; 2) defined by f j (z) = 1 z + a n+,j z n+ 1, we denote the Hadamard roduct (or convolution) of f 1 (z) and f 2 (z) by (f 1 f 2 )(z) = 1 z + a n+,1 a n+,2 z n+. Let us define the function φ (a, c; z) by φ (a, c; z) = 1 z + (a) n+1 (c) n+1 a nz n+ 1, for c 0, 1, 2,... and a C \ {0}, N = 1, 2, 3,... where (λ) n Pochhammer symbol. We note that is the where φ(a, c; z) = 1 z 2 F 1 (1, a, c; z), 2F 1 (1, a, c; z) = (1) n+1 (a) n+1 (c) n+1 is the well-known Gaussian hyergeometric function. Corresonding to the function φ (a, c; z), using the Hadamard roduct for f, we define a new linear oerator L (a, c) on by (1.1) L (a, c)f(z) = φ(a, c; z) f(z) = 1 z + z n n!, a n+z n+. The meromorhic functions with the generalized hyergeometric functions were considered recently by Dziok and Srivastava [12, 13], Liu [21], Liu and Srivastava [22 24], Cho and Kim [9], Ghanim and Darus [15]. For a function f, Albehbah and Darus [2] introduced and studied the differential oerator I k f(z) (1.2) I k f(z) = 1 z + P (n, k)a n z n, and by using (1.1) and (1.2), we define the oerator I k (L (a, c)f(z)). For a function f, we define I 0 (L (a, c)f(z)) = L (a, c)f(z),

3 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS 85 and for k = 1, 2, 3,..., we have [ ] 1 I k (L (a, c)f(z)) = z k z k 1 (Ik 1 L ( + k) (a, c)f(z)) + z (1.3) = 1 z + P (n +, k) a n+z n+, where P (n +, k) = n+!. (n+ k)! We note that I k (L 1(a, c)f) was studied earlier by Albehbah and Darus [1]. Also, it follows from (1.1) that (see [21]) (1.4) z((l (a, c)f(z))) = al (a + 1, c)f(z) (a + )L (a, c). Now, let 1 B < A 1 and for all z U, a function f is said to be a member of the class (A, B, k) a,c if it satisfies z(i k L (a, c)f(z)) + I k L (a, c)f(z) Bz(I k L (a, c)f(z)) + A(I k L (a, c)f(z) < 1. The class 1 (A, B, 0) a,c = (A, B) a,c was studied by Morga [27]. Note that, for a = c, = 1, (1 2µ, 1, k) a,c with 0 µ < 1, is the class introduced and studied in [1]. In Section 2, our first result concerns the coefficient estimates and distortion theorem for the class (A, B, k) a,c. 2. Coefficient Estimates and Distortion Theorems Our first result rovides a sufficient condition for a function f analytic in U, to be in the class (A, B, k) a,c. Theorem 2.1. Let the function f be defined by (1.1). If (2.1) P (n +, k) [(n + )(1 B) + (1 A)] a n+ (A B), where k N 0 = N {0}, 1 B < A 1, then f (A, B, k) a,c. Proof. To rove Theorem 2.1, we show that if f satisfies (2.1) then M(f, f ) = z(i k L (a, c)f(z)) + I k L (a, c)f(z) Bz(I k L (a, c)f(z)) + AI k L (a, c)f(z). Then for 0 < z = r < 1 we have M(f, f ) = P (n +, k) (c) n+2 (n + 2)a n+z n+ (A B) + P (n +, k) z ((n + )B + A)a n+z n+.

4 86 M. ALBEHBAH AND M. DARUS This gives (2.2) r M(f, f ) P (n +, k) (A B). [(n + )(1 B) + (1 A)] a n+ r n+2 The inequality in (2.2) holds true for all r(0 < r < 1). Therefore, by letting r 1 in (2.2), we have M(f, f ) P (n +, k) [(n + )(1 B) + (1 A)] a n+ (A B) 0, by the hyothesis (2.1). Hence it follows that z(i k L (a, c)f(z)) + I k L (a, c)f(z) Bz(I k L (a, c)f(z)) + AI k L (a, c)f(z), so that f (A, B, k) a,c. Our assertion in Theorem 2.1 is shar for function f of the form: (2.3) f n+ (z) = 1 z + (A B) P (n +, k)((n + )(1 B) + (1 A)) zn, where n 0 k N 0, 1 B < A 1. Corollary 2.1. Let the function f be defined by (1.1) and let f. (A, B, k) a,c. Then (2.4) a n+ (A B) P (n +, k)[(n + )(1 B) + (1 A)], where n 0, k N 0, 1 B < A 1. The result (2.4) is shar for function f n+ given by (2.3). Corollary 2.2. For k = 0, A = 1, and B = 1 in Theorem 2.1, we have (n + ) a n + 1 (c) n+2 and therefore the function L (a, c)f is starlike in U. Corollary 2.3. For k = 1, A = 1, and B = 1 in Theorem 2.1, we have (n + )2 a n + 1 (c) n+2 and therefore the function L (a, c)f is convex in U. If f The growth and distortion roerties for function f of the class (A, B, k) a,c are given in the following result.

5 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS 87 Theorem 2.2. If the function f be defined by (1.1) is in the class (A, B, k) a,c, then for z = r, 0 < r < 1, we have 1 r (A B) [2 (A + B)] r f(z) 1 r + (A B) [2 (A + B)] r and r (A B) +1 [2 (A + B)] r 1 f (z) r + (A B) +1 [2 (A + B)] r 1 with equality for f(z) = 1 z + (A B) [2 (A + B)] z. Proof. Since f (A, B, k) a,c, Theorem 2.1 readily yields the inequality (2.5) a (A B) n+ P (, k)[2 (A + B)]. and Thus, for 0 < z = r < 1, and making use of (2.5) we have f(z) 1 z + 1 r + r f(z) 1 z 1 r r a n+ z n+ a n+ 1 r + (A B) [2 (A + B)] r a n+ z n+ a n+ 1 r (A B) [2 (A + B)] r. Also from Theorem 2.1, it follows that (a) n+2 (n + ) a (A B) n+ P ( 1, k)[2 (A + B)], f (z) z +1 + (a) n+2 (n + ) a n+ z n+ 1 + r 1 r+1 (n + ) a n+ r +1 + (A B) [2 (A + B)] r 1

6 88 M. ALBEHBAH AND M. DARUS and f (z) z r 1 r+1 (n + ) a n+ z n+ 1 r + (A B) +1 [2 (A + B)] r 1. Thus, the roof of the theorem is comlete. (n + ) a n+ 3. Radii of Starlikeness and Convexity The radii of starlikeness and convexity for the class (A, B, k) a,c is given by the following theorems. Theorem 3.1. If the function f defined by (1.1) is in the class (A, B, k) a,c, then f is meromorhically starlike of order δ (0 δ 1) in the disk z < r 1 where r 1 = r 1 (A, B, k, δ) { } 1 P (n +, k)( δ)[(n + )(1 B) + (1 A)] n+2 = inf, n 0 (A B)(n + 3 δ) where the result is shar for the functions f n given by (2.3). Proof. It suffices to rove that z(l (a, c)f(z)) + L (a, c)f(z) δ, for z < r 1 we have z(l (a, c)f(z)) + L (a, c)f(z) = (3.1) (n + 2)an+ z n+ 1 + z a n+ z n+ (n + 2) an+ z n+2 1. an+ z n+2 Hence (3.1) holds true if ( (n + 2) a n+ z n+2 ( δ) 1 or (3.2) (n + 3 δ) a n+ z n+2 1. ( δ) ) a n+ z n+2

7 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS 89 With the aid of (2.1) and (3.2), it is true if (3.3) (n + 3 δ) z n+2 P (n, k)[(n + )(1 B) + (1 A)], ( δ) (A B) where n 0. Solving (3.3) with resect to z, we obtain { } 1 P (n +, k)( δ)[(n + )(1 B) + (1 A)] n+2 z <, (A B)(n + 3 δ) and this comletes the roof of Theorem 3.1. Theorem 3.2. If the function f defined by (1.1) is in the class (A, B, k) a,c, then f is meromorhically convex of order δ (0 δ 1) in the disk z < r 2 where r 2 = r 2 (A, B, k, δ) { } 1 P (n + 1, k)( δ)[(n + )(1 B) + (1 A)] n+2 = inf, n 0 (A B)(n + 3 δ) where the result is shar for the functions f n given by (2.3). Proof. By using the technique emloyed in the roof of Theorem 3.1, and with the aid of Theorem 2.1, we can show that z(l (a, c)f(z)) L (a, c)f(z) δ, for z < r 2. Thus we have the assertion of Theorem Convex Linear Combination Our next result involves a linear combination of several function of the tye (2.3). Theorem 4.1. Let (4.1) f 1 (z) = 1 z and (4.2) f n+ (z) = 1 z + (A B) P (n +, k)[(n + )(1 B) + (1 A)] zn+, for n 0 and k N 0. Then f (A, B, k) a,c if and only if it can exressed in the form (4.3) f(z) = λ +n 1 f +n 1 (z), where λ +n 1 0 and λ +n 1 = 1.

8 90 M. ALBEHBAH AND M. DARUS Proof. From (4.1), (4.2) and (4.3), it is easily seen that f(z) = λ +n 1 f +n 1 (z) Since = 1 z + =: 1 z + b n+ z n+. (A B) λ n+ P (n +, k)[(n + )(1 B) + (1 A)] (a) n+1 zn+ P (n +, k)[(n + )(1 B) + (1 B)] λ n+ (A B) (A B) P (n +, k)[(n + )(1 B) + (1 A)] (a) n+1 = λ n+ = 1 λ 1 0, it follows from Theorem 2.1 that the function f (A, B, k) a,c. Conversely, let us suose that f (A, B, k) a,c. Since Setting a n+ (A B) P (n +, k)[(n + )(1 B) + (1 A)] (a) n+1, n 0, k N 0. λ n+ = P (n +, k)[(n + )(1 B) + (1 B)] (A B) where n 0, k N 0, N and λ 1 = 1 λ n+, a n+, it follows that f(z) = λ +n 1f +n 1 (z). Theorem 4.2. The class (A, B, k) a,c is closed under convex linear combination. Proof. Suose that the function f 1 and f 2 defined by (4.4) f j (z) = 1 z + a n+j z n+, j = 1, 2; z U, are in the class (A, B, k) a,c. Setting f(z) = µf 1 (z) + (1 µ)f 2 (z), 0 µ 1,

9 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS 91 we find from (4.4) that f(z) = 1 z + { µa n+,1 + (1 µ)a n+,2 }z n+, 0 µ 1, N, z U. In view of Theorem 2.1, we have (P (n +, k)[(n + )(1 B) + (1 A)]) µ (P (n +, k)[(n + )(1 B) + (1 A)]) + (1 µ) { µa n+,1 + (1 µ)a n+,2 } (P (n +, k)[(n + )(1 B) + (1 A)]) µ(a B) + (1 µ)(a B) = (A B), a n+,1 a n+,2 which shows that f (A, B, k) a,c. Hence the theorem is comlete. 5. Inclusion Proerties We begin by recalling the following result (oularly known as Jack s Lemma), which we shall aly in roving our inclusion theorem (Theorem 5.1 below). Lemma 5.1 (Jack [18]). Let the (nonconstant) function ω(z) be analytic in U with ω(0) = 0. If ω(z) attains its maximum value on the circle z = r < 1 at a oint z 0 U, then z 0 ω (z 0 ) = γω(z 0 ), where γ is a real number and γ 1. Miller and Mocanu [25] gave roof to this lemma with the additional statement and made extensive use of it in their work on differential subordinations. Theorem 5.1. If then a (A B), 1 < B < A 1, N, B + 1 (A, B, k) a+1,c Proof. Let f (A, B, k) a+1,c and suose that (5.1) z(i k (L (a, c)f(z))) (I k L (a, c)f(z)) (A, B, k) a,c. = 1 + Aω(z) 1 + Bω(z),

10 92 M. ALBEHBAH AND M. DARUS where the function is either analytic or meromorhic in U, with ω(0) = 0. Then, by using (1.4) and (5.1), we have (5.2) a (Ik (L (a + 1, c)f(z))) (I k L (a, c)f(z)) = a + [ab (A B)]ω(z). 1 + Bω(z) Uon differentiating both sides of (5.2) with resect to z logarithmically, if we make use of (1.4) once again, we obtain (5.3) If we suose now that z(i k (L (a + 1, c)f(z))) (I k L (a + 1, c)f(z)) = 1 + Aω(z) 1 + Bω(z) (A B)zω (z) [1 + Bω(z)]{a + [ab (A B)]ω(z)}. and aly Jack s Lemma, we find that max ω(z) = ω(z 0) = 1, z 0 U, z z 0 z 0 ω (z 0 ) = γω(z 0 ), γ 1. Writing w(z 0 ) = e iθ (0 θ < 2π) and setting z = z 0 in (5.3), we get z 0 (I k L (a + 1, c)f(z)) + I k L 2 (a + 1, c)f(z 0 ) Bz 0 (I k L (a + 1, c)f(z 0 )) + A(I k L (a + 1, c)f(z 0 ) 1 = (a + γ) + [ab (A B)]e iθ 2 a + [ab γ (A B)]e iθ 1 2γ(1 + cosθ)[a(b + 1) (A B)] = a + [ab γ (A B)]e iθ 0, where a (A B), 1 < B < A 1, N, which obviously contradicts our B+1 hyothesis that f (A, B, k) a+1,c. Thus we must have ω(z) < 1, z U, and so, from (5.1), we conclude that f (A, B, k) a,c, which evidently comletes the roof of Theorem 5.1.

11 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS Remarks Hyergeometric functions are of secial interests among the comlex analysts, and their main interest are on the roerties and characterizations. The work has been around as early as in the 1900s. The classical results can be found in the books written by Exton [14], Bailey [6] and many other articles by many different authors, for examle: Carlson and Shaffer [8] and the recent one by Ghany [17]. Too many researchers have studied the geometrical roerties of this class and discovered many results. This tye of functions is then convolute with the meromorhic tye of functions to form a set of new class of functions. We mentioned before, the meromorhic functions with the generalized hyergeometric functions have been considered by many such as by Dziok and Srivastava [12, 13], Liu [21], Liu and Srivastava [22 24], Cho and Kim [9], and others (see [7, 10, 11, 30]). Motivated by their results, we defined the generalized hyergeometric functions with the differential oerator of meromorhic tye, and several interesting results are obtained. Other alications such as the Fekete-Szegö roblems, Hankel determinant and other coefficient related roblems are yet to be considered for this new class. Acknowledgements. This work was suorted by MOHE: FRGS/1/2016/STG06/ UKM/01/1. References [1] M. Albehbah and M. Darus, On new subclasses of meromorhic univalent functions with ositive coefficients, Far East J. Math. Sci. 97 (4) (2015), [2] M. Albehbah and M. Darus, Subclasses of meromorhic multivalent functions, Acta Univ. Aul. Math. Inform. 43 (2015), [3] M. K. Aouf, New criteria for multivalent meromorhic starlike functions of order α, Proc. Jaan. Acad. Ser. A. Math. Sci. 69 (1993), [4] M. K. Aouf and H. M. Hossen, New criteria for meromorhic -valent starlike functions, Tsukuba J. Math. 8 (17) (1993), [5] M. K. Aouf and H. M. Srivastava, A new criterion for meromorhically -valent convex functions of order alha, Math. Sci. Res. Hot-Line 8 (2) (1997), [6] W. N. Bailey, Basic hyergeometric series, Ch. 8 in Generalised Hyergeometric Series, Cambridge Univ. Press, , [7] S. K. Bajai,A note on a class of meromorhic univalent functions, Rev. Roumaine Math. Pures Al. 22 (3) (1977), [8] B. C. Carlson and D. B. Shaffer, Starlike and restarlike hyergeometric functions, SIAM J. Math. Anal. 15 (1984), [9] N. E. Cho and I. H. Kim, Inclusion roerties of certain classes of meromorhic functions associated with the generalized hyergeometric function, Al. Math. Comut. 187 (2007), [10] N. E. Cho, S. H. Lee and S. Owa, A class of meromorhic univalent functions with ositive coefficients, Kobe J. Math. 4 (1) (1987), [11] Y. Dinggong, On a classes of meromorhic starlike multivalent functions, Bull. Inst. Math. Acad. Sin. 24 (1996),

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13 NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS 95 1 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600, Selangor D. Ehsan, Malaysia address: malbehbah@yahoo.com School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600, Selangor D. Ehsan, Malaysia address: maslina@ukm.edu.my