2 The Main result

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

meromorphic functions ^∗

H. Irmak, G. Tınaztepe, N. Tuneski & M. S ¸an

Abstract

In the present work, using an ordinary differential operator of orderq(q∈ N⁰ :={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

LetM(p) denote the class of functions f of the following form f(z) =z^−p+

∞

X

k=p+1

akz^k (p∈N:=N0\ {0};ak∈C), (1) which areanalytic andmultivalently meromorphic in thepunctured unit disk

D:={z : z∈Cand 0<|z|<1}=U− {0}, where Cdenotes the set of complex numbers.

Also letMS(p;α) and MC(p;α) be the well-known subclasses of the class M(p) consisting, respectively, of functions which aremultivalently meromorphic starlike of order αandmultivalently meromorphic convex of orderαinD,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 variablez,one easily obtains the following (ordinary) differential operator

f^(q)(z) =(p+q−1)!

(p−1)! (−1)^qz^−p−q+

∞

X

k=p+1

k!

(k−q)!akz^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 Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted May, 2009. Published June, 2009.

17

where f ∈ M(p), p > q, p∈N,andq∈N0:=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 functionsf satisfying the following inequality:

<e

zF⁰(z) F(z)

<−α (z∈D; 0≤α < p+q;p∈N;q∈N0), where, here and throughout this paper, the above functionF is defined by

F(z) = (1−λ)f^(q)(z) +λzf^(1+q)(z) (0≤λ≤1;p > q;p∈N;q∈N0). (3) One can note that by choosing specific values of p, q and/or λwe receive some well known classes of multivalently meromorphic functions. Namely,

Ω^λ₀(p;α) =:Vλ(p;α) (0≤λ≤1; 0≤α < p;p∈N), Ω^λ₀(1;α) =:Wλ(α) (0≤λ≤1; 0≤α <1),

Ω⁰_q(p;α) =:Aq(p;α) (0≤α < p+q;p∈N;q∈N0), Ω¹_q(p;α) =:Bq(p;α) (0≤α < p+q;p∈N;q∈N0), Ω⁰₀(p;α)≡ S(p;α) (0≤α <1),

Ω¹₀(p;α)≡ K(p;α) (0≤α <1), Ω⁰₀(1;α)≡ S(α) (0≤α <1), Ω¹₀(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 functionwbe non-constant and analytic inUwithw(0) = 0. If |w(z)| attains its maximum value on the circle |z| =r < 1 at a point z₀, thenz₀w⁰(z₀) =cw(z₀), where c is real number andc≥1.

2 The Main result

Theorem 2.1 Let the functionsf andFbe defined by (1) and (3), respectively.

Also let the functionHbe defined by H(z) :=

1 +q+zF⁰⁰(z) F⁰(z) +p

·

q+zF⁰(z) F(z) +p

−1

(z∈D). (4) If Hsatisfies

<e{H(z)}>1−β (5)

for all z∈D, thenf ∈Ω^λ_q(p;α) and

<en

[H(z)]⁻¹o

<(1−β)⁻¹ (z∈D), (6) where β:= [2(p+q)−α]⁻¹ and 0≤α < p+q.

Proof.

Letf be of the form (1) Then, in view of (3), one easily obtains that zF⁰(z)

F(z) =(p+q)φ(q, λ;p)(−1)^1+q+P∞

k=p+1(k−q)ψ(q, λ;k)akz^k+p φ(q, λ;p)(−1)^q+P∞

k=p+1ψ(q, λ;k)a_kz^k+p ,

where

φ(q, λ;p) := (p+q−1)![1−λ(p+q+ 1)]

(p−1)!

and

ψ(q, λ;k) := k![1 +λ(k−q−1)]

(k−q)! , (k≥p+ 1;p > q;p∈N;q∈N0).

Now let us define a functionw(z) with

−q−zF⁰(z)

F(z) −p= (p+q−α)w(z) (w(z)6= 0). (7) It is clear that the function w is both analytic inU withw(0) = 0 and mero- morphic inD.We also find from (7) that

−1−q−zF⁰⁰(z) F⁰(z) −p

= (p+q−α)w(z)

1−zw⁰(z)

w(z) · 1

p+q+ (p+q−α)w(z)

. (8)

By using (7) and (8), we easily arrive at G(z) := 1− H(z) =zw⁰(z)

w(z) · 1

p+q+ (p+q−α)w(z), (9) where His defined by (4).

Now let suppose that there exists a point z₀ ∈ U such that max{|w(z)| :

|z| ≤ |z0|} = |w(z0)| = 1 and w(z₀) = e^iθ (0 ≤ θ < 2π). By applying Jack lemma we then havez₀w⁰(z₀) =cw(z₀) (c≥1).Thus, in view of (9), we obtain

<e{G(z0)}=c<e[(p+q+ (p+q−α)e^iθ)⁻¹]≥[2(p+q)−α]⁻¹=β or, equivalently,

<e{H(z0)} ≤1−β,

which is a contradiction to (5), where β is given by in the statement of the Theorem 2.1. Hence, we conclude that |w(z)| < 1 for all z in U, and the definition (7) immediately yields the inequality

q+zF⁰(z) F(z) +p

< p+q−α, which implies

<e

−zF⁰(z) F(z)

> α

(z∈D; 0≤α < p+q;p∈N;q∈N0), that isf ∈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 1.2.

3 Certain consequences of the main result

As we indicated in the Section 1, i.e., by fixing some specific admissible values of parametersp, qand/orλ, from Theorem 2.1 we easily receive many interest- ing results concerning the functionsf in the classesVλ(p;α),Wλ(α),Aq(p;α), Bq(p;α),S(p;α),K(p;α),S(α) and alsoK(α).Here we only state some of them as corollaries.

By takingq= 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(z) = (1−λ)f(z) +λzf⁰(z).If

<e





1 +^zF_F0⁰⁰(z)^(z)+p

zF⁰(z) F(z) +p



>1−β for allz∈D,then f ∈ Vλ(p;α) and

<e





zF⁰(z) F(z) +p 1 +^zF_F0⁰⁰(z)^(z)+p



< 1

1−β (z∈D), whereβ := 1/(2p−α).

By settingq= 0 and λ= 0 in Theorem 2.1, we receive Corollary 3.2 Let f ∈ M(p), p∈N, and0≤α < p. If

<e





1 +^zf_f0⁰⁰(z)^(z)+p

zf⁰(z) f(z) +p



>1−β

for all z∈D, thenf ∈ S(p;α)and

<e





zf⁰(z) f(z) +p 1 + ^zf_f0⁰⁰(z)^(z)+p



< 1

1−β (z∈D), where β:= 1/(2p−α).

By lettingq= 0 andλ= 1 in Theorem 2.1, we have Corollary 3.3 Let f ∈ M(p), p∈N, and0≤α < p. If

<e





1 + ^z(zf_(zf0⁰(z))^(z))000 +p

(zf⁰(z))⁰ f⁰(z) +p



>1−β for all z∈D, thenf(z)∈ K(p;α)and

<e





(zf⁰(z))⁰ f⁰(z) +p 1 + ^z(zf_(zf0⁰(z))^(z))000 +p



< 1

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.17-1383/1) and The Scientific and Technical Research Council of Turkey (TUBITAK) (Project No. TBGA- U-105T056).

References

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[3] A. W. Goodman,Univalent Functions, Vols.IandII, Polygonal Publishing House, Washington, New Jersey, 1983.

[4] H. Irmak, N. E. Cho, ¨O. F. C¸ etin and R. K. Raina, Certain inequalities in- volving meromorphically multivalent functions,Hacet. Bull. Nat. Sci. Eng.

Ser. B30(2001), 39-43.

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[6] I. S. Jack, Functions starlike and convex of orderα,J. London Math. Soc.

3(1971), 469-474.

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H. Irmak & M. S¸an Department of Mathematics Faculty of Science and Letters C¸ ankırı Karatekin University Tr-18100, C¸ ankırı, Turkey

e-mails: hirmak70@gmail.com, mufitsan@hotmail.com N. Tuneski

Faculty of Mechanical Engineering

Karpoˇs II b.b., 1000 Skopje, Republic of Macedonia e-mail: nikolat@mf.ukim.edu.mk

G. Tınaztepe

Vocational School of Technical Sciences Akdeniz University

Tr-07058, Antalya, Turkey e-mail: gtinaztepe@akdeniz.edu.tr