Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

Applied Mathematics and Computation 218 (2011) 817–821

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Ordinary differential operator and some of its applications to certain meromorphically p-valent functions M. Sß an, H. Irmak ⇑ Department of Mathematics, Faculty of Science, Çankırı Karatekin University, Tr-18100, Çankırı, Turkey

a r t i c l e

i n f o

a b s t r a c t

Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Unit open disc Punctured open unit disc Analytic and meromorphically p-valent function Strongly starlikeness Strongly convexity Inequalities Ordinary differential operator

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction, definitions and motivation Let us denote by MðpÞ the class of functions f(z) of the form:

f ðzÞ ¼ ap zp þ

1 X

ak zk ;

ðap – 0; ak 2 C; k ¼ 1  p; p 2 N :¼ f1; 2; 3; . . .gÞ;

ð1:1Þ

k¼1p

which are analytic and meromrophically p-valent in the punctured open unit disc D ¼ U  f0g, where U :¼ fz : z 2 C and jzj < 1g, and also let M :¼ Mð1Þ. A function f ðzÞ 2 MðpÞ is said to be p-valently meromorphic starlike of order a in U if it is satisfies the inequality:

  0 zf ðzÞ > a ð0 6 a < p; p 2 NÞ: Re  f ðzÞ

ð1:2Þ

A function f(z) belonging to the class MðpÞ is said to be p-valently meromorphic convex of order a in U if it is also satisfies the inequality:

  00 zf ðzÞ Re 1  0 > a ð0 6 a < p; p 2 NÞ: f ðzÞ

ð1:3Þ

It is obviously that the rational type functions in (1.2) and (1.3) are both analytic in D and meromorphic in U. In view of the related two definitions, a function f(z) in the general class MðpÞ is p-valently meromorphic starlike of order a ð0 6 a < p; p 2 NÞ if and only if zf0 (z)/p is p-valently meromorphic convex of order a ð0 6 a < p; p 2 NÞ in the disc U (and also in DÞ. For their details, see [3,4]. ⇑ Corresponding author. E-mail addresses: mufi[email protected] (M. Sßan), [email protected] (H. Irmak). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.037

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M. Sßan, H. Irmak / Applied Mathematics and Computation 218 (2011) 817–821

The main purpose of the present paper is to reveal certain properties of functions belonging to the subclasses denoted by MS p ðq; aÞ and MC p ðq; aÞ of the general class MðpÞ, which consist of p-valently meromorphic starlike and p-valently meromorphic convex functions of order a ð0 6 a < p; p 2 NÞ, respectively. Indeed, we have

! ) ð1þqÞ zf ðzÞ >a f ðzÞ 2 MðpÞ : Re f ðqÞ ðzÞ

( MS p ðq; aÞ :¼

ð1:4Þ

and

(

ð2þqÞ

zf ðzÞ f ðzÞ 2 MðpÞ : Re 1 þ ð1þqÞ f ðzÞ

MCp ðq; aÞ :¼

)

!

>a ;

ð1:5Þ

where z 2 U and, for each f ðzÞ 2 MðpÞ and for all q 2 N0 :¼ N [ f0g, the operator f(q)(z) is defined by:

f ðqÞ ðzÞ ¼

1 1 X X ðk þ q  1Þ! j! ð1Þq ak zkq þ aj zjq : ðk  1Þ! ðj  qÞ! j¼q k¼p

ð1:6Þ

Various special cases of the classes defined in (1.4) and (1.5) were studied by many earlier researchers on the topic of both analytic and geometric function theory. As we know, for instance, by taking q = 0 and p = 1 in the definitions in (1.4) and (1.5), the well-known subclasses MCðaÞ :¼ MC1 ð1; aÞ and MS  ðaÞ :¼ MS 1 ð1; aÞ, which are the functions classes of meromorphically convex and meromorphically starlike of order a (0 6 a < 1) in the related disc, are respectively obtained. For the certain earlier results consisting the functions belonging to the (general) class MðpÞ and also certain differential operator, the papers in [1,2,5–12] can be checked. We note also that Nunokawa et al. [15] showed that

f ðzÞ 2 MC





að3  2aÞ ) f ðzÞ 2 MS  ðaÞ; 2ð1  aÞ

where for all a < 0 and f ðzÞ 2 M. We need the following lemmas for the proofs of the main results. Lemma 1.1 [13]. Let / ¼ /ðu; v Þ : D  C2 ! C be a complex valued function, and also let u = u1 + iu2 and that the function /(u, v) satisfies the following conditions: (i) /(u, v) is continuous, (ii) ð1; 0Þ 2 D and Ref/ð1; 0Þg > 0, and (iii) Ref/ðiu2 ; v 1 Þg 6 0 for all (iu2, v1) such that

v = v1 + iv2. Suppose

2 2

v 1 6  1þu2 .

Let a function P(z) = 1 + c1z + c2z2 +    be regular in U such that ðPðzÞ; zP 0 ðzÞÞ 2 D for all z 2 U. Then

Ref/ðPðzÞ; zP 0 ðzÞÞg > 0 ) RefPðzÞg > 0 ðz 2 UÞ: Lemma 1.2 [14]. Let the function Q(z) be analytic in U with Q(0) = 1 and Q ðzÞ – 0 ðz 2 UÞ. Suppose also there exists a point z0 2 U such that

jargfQ ðzÞgj <

pb 2

ðjzj < jz0 jÞ

and

jargfQ ðz0 Þgj ¼

pb 2

ðb > 0Þ:

Then we have

z0 Q 0 ðz0 Þ ¼ ijbQðz0 Þ; where

jP

  1 1 pb aþ when argfQ ðz0 Þg ¼ ; 2 a 2

j6

  1 1 pb aþ when argfQ ðz0 Þg ¼  ; 2 a 2

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M. Sßan, H. Irmak / Applied Mathematics and Computation 218 (2011) 817–821

and 1

½Qðz0 Þb ¼ ia ða > 0Þ: 2. The main results

Theorem 2.1. Let 0 6 a < p, p 2 N, q 2 N0 and also let f ðzÞ 2 MðpÞ. If

(

ð2þqÞ

Re  1 þ

zf ðzÞ f ð1þqÞ ðzÞ

!)

> a ðor f ðzÞ 2 MC p ðq; aÞÞ;

ð2:1Þ

then

(

!) ð1þqÞ zf ðzÞ Re  > a ðor f ðzÞ 2 MS p ðq; aÞÞ: f ðqÞ ðzÞ

ð2:2Þ

Proof. Define P(z) by ð1þqÞ

zf ðzÞ ¼ a  ðp þ q þ aÞPðzÞ: f ðqÞ ðzÞ

ð2:3Þ

Then, it is easily seen that the function P(z) = 1 + c1z + c2z2 +    is both analytic in D and meromorphic in U. By differentiating the both sides of (2.3) logarithmically, we find that ð2þqÞ

zf ðzÞ  1 þ ð1þqÞ f ðzÞ

!

ð1þqÞ

¼

zf ðzÞ ðp þ q þ aÞzP 0 ðzÞ ðp þ q þ aÞzP 0 ðzÞ a þ ðp þ q þ a ÞPðzÞ þ þ ¼  : f ðqÞ ðzÞ a  ðp þ q þ aÞPðzÞ a  ðp þ q þ aÞPðzÞ

Since

(

ð2þqÞ

zf ðzÞ Re  1 þ ð1þqÞ f ðzÞ

!) > a;

it is easily received that

! ) ð2þqÞ zf ðzÞ ðp þ q þ aÞzP 0 ðzÞ Re  1 þ ð1þqÞ > 0:  a ¼ 2a þ ðp þ q þ aÞPðzÞ þ f ðzÞ a  ðp þ q þ aÞPðzÞ (

Let us next define the function /(u, v) as in the form:

/ðu; v Þ ¼ 2a þ ðp þ q þ aÞu þ

ðp þ q þ aÞv : a  ðp þ q þ aÞu

Then, /(u, v) satisfies the conditions of Lemma 1.1, indeed, we get: a gÞ  C, (i) /(u, v) is continuous in D ¼ ðC  fpþqþ a (ii) ð1; 0Þ 2 D and Reð/ð1; 0ÞÞ ¼ p  a þ q > q P 0,

(iii) for all (iu2, v1) such that

2 2

v 1 6  1þu2 ,

 Ref/ðiu2 ; v 1 Þg ¼ Re 2a þ ðp þ q þ aÞiu2 þ ¼ 2a þ

ðp þ q þ aÞv 1 a  ðp þ q þ aÞiu2



 ¼ 2a þ Re

ðp þ q þ aÞv 1 a  ðp þ q þ aÞiu2



aðp þ q þ aÞv 1 aðp þ q þ aÞð1 þ u22 Þ 6 2a  6 0; 2 2 2ða2 þ ðp þ q þ aÞ2 u22 Þ a þ ðp þ q þ aÞ u2 2

which yields the inequality RefPðzÞg > 0 for all z in U. From the definition of the function P(z) defined as in (2.3), the following assertion:

( Re 

!) ð1þqÞ zf ðzÞ > a f ðqÞ ðzÞ

is clearly obtained. Therefore, this completes the proof of Theorem 2.1. h If we first put q = 0 in Theorem 2.1, we get the following result.

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M. Sßan, H. Irmak / Applied Mathematics and Computation 218 (2011) 817–821

Corollary 2.1. Let f ðzÞ 2 MðpÞ. If

   00 zf ðzÞ > a ðor f ðzÞ 2 MC p ð0; aÞÞ; Re  1 þ 0 f ðzÞ then

  0 zf ðzÞ > a ðor f ðzÞ 2 MS p ð0; aÞÞ: Re  f ðzÞ If we take q = 0 and a ? 0+ in Theorem 2.1, we then receive the following result obtained by Nunokawa [15]. Corollary 2.2. Let f ðzÞ 2 M. If f 2 MCð0Þ, then f ðzÞ 2 MS  ð0Þ. Theorem 2.2. Let 0 < b < 1, z 2 U and also let f ðzÞ 2 MðpÞ. If

    ð2þqÞ  zf ð1þqÞ ðzÞ zf ðzÞ     1 þ ð1þqÞ  < b   ðqÞ   f ðzÞ  f ðzÞ 

ð2:4Þ

is satisfied, then

 ! ð1þqÞ  1 zf ðzÞ  pb   : < arg   p þ q f ðqÞ ðzÞ  2

ð2:5Þ

Proof. We suppose that the inequality:

    ð2þqÞ  zf ð1þqÞ ðzÞ zf ðzÞ     1 þ ð1þqÞ  < b   ðqÞ   f ðzÞ  f ðzÞ  holds for all z in U. Take Q(z) as in the form: ð1þqÞ

QðzÞ ¼ 

1 zf ðzÞ  : p þ q f ðqÞ ðzÞ

ð2:6Þ

Then, it is easily seen that Q(z) is both an analytic function in D with Q(0) = 1 and meromorphic function in U. The definition (2.6) yields that

  0   ð2þqÞ ð1þqÞ  zQ ðzÞ  zf ðzÞ zf ðzÞ     Q ðzÞ  ¼ 1 þ f ð1þqÞ ðzÞ  f ðqÞ ðzÞ ;

ð2:7Þ

and by applying Lemma 1.2 to the equality (2.7), we then get that

    zQ 0 ðzÞ       ¼ jijbj P b a þ 1 P b;    Q ðzÞ z¼z0  2 a or, equivalently,

   ð1þqÞ  ð2þqÞ  z0 f ðz0 Þ ðz0 Þ 1 þ z0 f  P b    f ðqÞ ðz Þ : f ð1þqÞ ðz0 Þ  0

ð2:8Þ

But, the result in (2.8) is a contradiction with the hypothesis of the related theorem. Hence, in view of Lemma 1.2 and the definition of Q(z) defined by (2.6) immediately yields that

 ( ) ð1þqÞ  1 zf ðzÞ  pb   ðqÞ ; jarg fQ ðzÞgj ¼ arg  <  p þ q f ðzÞ  2 which is the desired inequality. As in Theorem 2.1, Theorem 2.2 also includes several results which will be interesting and/or important (for example; strongly starlikeness, strongly convexity, starlikeness and also convexity) for both analytic and geometric function theory, by choosing suitable values of the parameter q 2 N0 . These obvious considerations are omitted in this work.

M. Sßan, H. Irmak / Applied Mathematics and Computation 218 (2011) 817–821

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Acknowledgments _ The work in this investigation was supported by both TÜBITAK (The Scientific and Technological Research Council of Turkey) under the Project Number TBAG-U/132(105T056) and Çankırı Karatekin University (Çankırı, Turkey). References [1] M.P. Chen, H. Irmak, H.M. Srivastava, Some families of multivalently analytic functions with negative coefficients, J. Math. Anal. Appl. 214 (2) (1997) 674–690. [2] M.P. Chen, H. Irmak, H.M. Srivastava, Some multivalent functions with negative coefficients defined by using a differential operator, Panamer. Math. J. 6 (2) (1996) 55–64. [3] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Bd. 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo. [4] A.W. Goodman, Univalent Functions, vols. I and II, Polygonal Publishing Company, Washington, New Jersey, 1983. [5] H. Irmak, G. Tınaztepe, N. Tuneski, M. Sßan, An ordinary differential operator and its applications to certain classes of multivalently meromorphic functions, Bull. Math. Anal. Appl. 1 (2) (2009) 17–22. [6] H. Irmak, Ö.F. Çetin, Some theorems involving inequalities on p-valent functions, Turkish J. Math. 23 (3) (1999) 453–459. [7] H. Irmak, R.K. Raina, Certain properties arising from some inequalities concerning subclasses of multivalently analytic functions, Math. Inequal. Appl. 10 (2) (2007) 327–334. [8] H. Irmak, R.K. Raina, On certain classes of functions associated with multivalently analytic and multivalently meromorphic functions, Soochow J. Math. 32 (3) (2006) 413–419. [9] H. Irmak, S. Owa, Certain inequalities for multivalent starlike and meromorphically multivalent starlike functions, Bull. Inst. Math. Acad. Sinica 31 (1) (2003) 11–21. [10] H. Irmak, R.K. Raina, New classes of non-normalized meromorphically multivalent functions, Sarajevo J. Math. 3(16) (2) (2007) 157–162. [11] H. Irmak, N.E. Cho, A differential operator and its applications to certain multivalently analytic functions, Hacet. J. Math. Stat. 36 (1) (2007) 1–6. [12] H. Irmak, Some applications of Hadamard convolution to multivalently analytic and multivalently meromorphic functions, Appl. Math. Comput. 187 (1) (2007) 207–214. [13] S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978) 289–305. [14] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (7) (1993) 234–237. [15] M. Nunokawa, O.P. Ahuja, On meromorphic starlike and convex function, Indian J. Pure Appl. Math. 32 (7) (2001) 1027–1032.