An ordinary differentail operator and its applications to certain classes of multivalently meromorphic functions

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1 Bulletin of Mathematical analysis and Applications ISSN: , URL: Volume 1, Issue 2, (2009), Pages An ordinary differentail operator and its applications to certain classes of multivalently meromorphic functions H. Irmak, G. Tınaztepe, N. Tuneski & M. Şan Abstract In the present work, using an ordinary differential operator of order q (q N 0 := {0, 1, 2, }), a general class of meromorphic functions which are analytic and multivalent in the punctured unit disk is firstly introduced. Sufficient condition for a function in the related class is then obtained. Several useful consequences of the main results are also pointed out. 1 Introduction and Definitions Let M(p) denote the class of functions f of the following form f = z p + k=p+1 a k z k (p N := N 0 \ {0}; a k C), (1) which are analytic and multivalently meromorphic in the punctured unit disk D := {z : z C and 0 < z < 1} = U {0}, where C denotes the set of complex numbers. Also let MS(p; α) and MC(p; α) be the well-known subclasses of the class M(p) consisting, respectively, of functions which are multivalently meromorphic starlike of order α and multivalently meromorphic convex of order α in D, where 0 α < p (p N). (See [2], [3] and [7] for further details). Upon differentiating both sides of (1), q-times with respect to the complex variable z, one easily obtains the following (ordinary) differential operator f (q) = (p + q 1)! (p 1)! ( 1) q z p q + k=p+1 k! (k q)! a kz k q, (2) Mathematics Subject Classifications: 30C45. Key words: Multivalently meromrophic function, multivalently meromrophic starlikeness, multivalently meromrophic convexity, ordinary differential operator, differential inequalities, Jack s lemma. c 2009 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted May, Published June,

2 18 Ordinary differentail operator and its certain applications where f M(p), p > q, p N, and q N 0 := N {0}. The operator defined by (2) has been studied earlier by several researchers (see, for example, [1], [4] and [5]). Using the ordinary differential operator defined in (2), we now introduce a general subclass Ω λ q (p; α) of the class of multivalently meromorphic functions M(p), which consists of functions f satisfying the following inequality: ( zf ) Re < α (z D; 0 α < p + q; p N; q N 0 ), F where, here and throughout this paper, the above function F is defined by F = (1 λ)f (q) + λzf (1+q) (0 λ 1; p > q; p N; q N 0 ). (3) One can note that by choosing specific values of p, q and/or λ we receive some well known classes of multivalently meromorphic functions. Namely, Ω λ 0 (p; α) =: V λ (p; α) (0 λ 1; 0 α < p; p N), Ω λ 0 (1; α) =: W λ (α) (0 λ 1; 0 α < 1), Ω 0 q(p; α) =: A q (p; α) (0 α < p + q; p N; q N 0 ), Ω 1 q(p; α) =: B q (p; α) (0 α < p + q; p N; q N 0 ), Ω 0 0(p; α) S(p; α) (0 α < 1), Ω 1 0(p; α) K(p; α) (0 α < 1), Ω 0 0(1; α) S(α) (0 α < 1), Ω 1 0(1; α) K(α) (0 α < 1). In this investigation we obtain sufficient condition for a function f M(p) to be in Ω λ q (p; α). In addition we give several corollaries of the main result. For that purpose we will use the method of the differential inequalities and the well-known assertion of Jack [6]. Lemma 1.1 Let the function w be non-constant and analytic in U with w(0) = 0. If w attains its maximum value on the circle z = r < 1 at a point z 0, then z 0 w (z 0 ) = cw(z 0 ), where c is real number and c 1. 2 The Main result Theorem 2.1 Let the functions f and F be defined by (1) and (3), respectively. Also let the function H be defined by ( H := 1 + q + zf ) ( F q + zf ) 1 F (z D). (4) If H satisfies Re {H} > 1 β (5)

3 H. Irmak, G. Tınaztepe, N. Tuneski & M. Şan 19 for all z D, then f Ω λ q (p; α) and { Re [H] 1} < (1 β) 1 (z D), (6) where β := [2(p + q) α] 1 and 0 α < p + q. Proof. Let f be of the form (1) Then, in view of (3), one easily obtains that where and zf F = (p + q)φ(q, λ; p)( 1)1+q + k=p+1 (k q)ψ(q, λ; k)a kz k+p φ(q, λ; p)( 1) q + k=p+1 ψ(q, λ; k)a, kz k+p φ(q, λ; p) := ψ(q, λ; k) := (p + q 1)![1 λ(p + q + 1)] (p 1)! k![1 + λ(k q 1)], (k q)! (k p + 1; p > q; p N; q N 0 ). Now let us define a function w with q zf F p = (p + q α)w (w 0). (7) It is clear that the function w is both analytic in U with w(0) = 0 and meromorphic in D. We also find from (7) that 1 q zf F p = (p + q α)w [ ] 1 zw w 1. p + q + (p + q α)w (8) By using (7) and (8), we easily arrive at G := 1 H = zw w 1 p + q + (p + q α)w, (9) where H is defined by (4). Now let suppose that there exists a point z 0 U such that max{ w : z z 0 } = w(z 0 ) = 1 and w(z 0 ) = e iθ (0 θ < 2π). By applying Jack lemma we then have z 0 w (z 0 ) = cw(z 0 ) (c 1). Thus, in view of (9), we obtain Re {G(z 0 )} = c Re[(p + q + (p + q α)e iθ ) 1 ] [2(p + q) α] 1 = β or, equivalently, Re {H(z 0 )} 1 β,

4 20 Ordinary differentail operator and its certain applications which is a contradiction to (5), where β is given by in the statement of the Theorem 2.1. Hence, we conclude that w < 1 for all z in U, and the definition (7) immediately yields the inequality q + zf F < p + q α, which implies ( Re zf ) > α F (z D; 0 α < p + q; p N; q N 0 ), that is f Ω λ q (p; α). At the end (5) implies (6) since 1 β > 0 because of 2(p + q) α > 1. This completes the proof of the Theorem Certain consequences of the main result As we indicated in the Section 1, i.e., by fixing some specific admissible values of parameters p, q and/or λ, from Theorem 2.1 we easily receive many interesting results concerning the functions f in the classes V λ (p; α), W λ (α), A q (p; α), B q (p; α), S(p; α), K(p; α), S(α) and also K(α). Here we only state some of them as corollaries. By taking q = 0 in Theorem 2.1, we first obtain the following corollary. Corollary 3.1 Let f M(p), p N, 0 α < p, 0 λ 1, and also let F = (1 λ)f + λzf. If Re 1 + zf F > 1 β zf F for all z D, then f V λ (p; α) and zf F Re < zf F 1 β (z D), where β := 1/(2p α). By setting q = 0 and λ = 0 in Theorem 2.1, we receive Corollary 3.2 Let f M(p), p N, and 0 α < p. If Re 1 + zf f > 1 β zf f

5 H. Irmak, G. Tınaztepe, N. Tuneski & M. Şan 21 for all z D, then f S(p; α) and zf f Re < zf f 1 β (z D), where β := 1/(2p α). By letting q = 0 and λ = 1 in Theorem 2.1, we have Corollary 3.3 Let f M(p), p N, and 0 α < p. If Re 1 + z(zf ) (zf ) > 1 β (zf ) f for all z D, then f K(p; α) and (zf ) f Re < z(zf ) (zf ) 1 β (z D), where β := 1/(2p α). Acknowledgements The work on this paper was supported by the Joint Research Project financed by The Ministry of Education and Science of the Republic of Macedonia (MESRM) (Project No /1) and The Scientific and Technical Research Council of Turkey (TUBITAK) (Project No. TBGA- U-105T056). References [1] M. P. Chen, H. Irmak and H. M. Srivastava, Some families of multivalently analytic functions with negative coefficients, J. Math. Anal. Appl. 214 (2) (1997), [2] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, [3] A. W. Goodman, Univalent Functions, Vols. I and II, Polygonal Publishing House, Washington, New Jersey, [4] H. Irmak, N. E. Cho, Ö. F. Çetin and R. K. Raina, Certain inequalities involving meromorphically multivalent functions, Hacet. Bull. Nat. Sci. Eng. Ser. B 30 (2001),

6 22 Ordinary differentail operator and its certain applications [5] H. Irmak and N. E. Cho, A differential operator and its application to certain multivalently analytic functions, Hacet. J. Math. Stat. 36 (1) (2007), 1-6. [6] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), [7] H. M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, H. Irmak & M. Şan Department of Mathematics Faculty of Science and Letters Çankırı Karatekin University Tr-18100, Çankırı, Turkey s: hirmak70@gmail.com, mufitsan@hotmail.com N. Tuneski Faculty of Mechanical Engineering Karpoš II b.b., 1000 Skopje, Republic of Macedonia nikolat@mf.ukim.edu.mk G. Tınaztepe Vocational School of Technical Sciences Akdeniz University Tr-07058, Antalya, Turkey gtinaztepe@akdeniz.edu.tr