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# Majorization properties for certain new classes of analytic functions using the Salagean operator

- Shu-Hai Li
^{1}Email author, - Huo Tang
^{1, 2}and - En Ao
^{1}

**2013**:86

https://doi.org/10.1186/1029-242X-2013-86

© Li et al.; licensee Springer 2013

**Received:**9 November 2012**Accepted:**14 February 2013**Published:**4 March 2013

## Abstract

In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Salagean operator. Moreover, we point out some new and interesting consequences of our main result.

**MSC:**30C45.

## Keywords

- analytic functions
- multivalent functions
*α*-uniformly starlike functions of order*β**α*-uniformly convex functions of order*β*- subordination
- majorization property

## 1 Introduction and definitions

*f*and

*g*be two analytic functions in the open unit disk

*f*is majorized by

*g*in Δ (see [1]) and write

*φ*, analytic in Δ, such that

It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions.

*f*and

*g*, analytic in Δ, we say that the function

*f*is subordinate to

*g*in Δ if there exists a Schwarz function

*ω*, which is analytic in Δ with

*g*is univalent in Δ, then

that are analytic and *p*-valent in the open unit disk Δ. Also, let ${A}_{1}=A$.

*q*th-order ordinary differential operator by

In view of (1.6), it is clear that ${D}^{0}{f}^{(0)}(z)=f(z)$, ${D}^{0}{f}^{(1)}(z)=z{f}^{\prime}(z)$ and ${D}^{m}{f}^{(0)}(z)={D}^{m}f(z)$ is a known operator introduced by Salagean [3].

**Definition 1.1**A function $f(z)\in {A}_{p}$ is said to be in the class ${L}_{p,q}^{j,l}[A,B;\alpha ,\gamma ]$ of

*p*-valent functions of complex order $\gamma \ne 0$ in Δ if and only if

- (1)
${L}_{p,q}^{j,l}[A,B;0,\gamma ]={S}_{p,q}^{j,l}[A,B;\gamma ]$;

- (2)
${L}_{1,0}^{m,n}[A,B;\alpha ,1]={U}_{m,n}(\alpha ,A,B)$;

- (3)
${L}_{1,0}^{1,0}[1-2\beta ,-1;\alpha ,1]=US(\alpha ,\beta )$ ($0\le \beta <1$) (

*α*-uniformly starlike functions of order*β*); - (4)
${L}_{2,1}^{1,0}[1-2\beta ,-1;\alpha ,1]=UK(\alpha ,\beta )$ ($0\le \beta <1$) (

*α*-uniformly convex functions of order*β*); - (5)
${L}_{p,0}^{n+1,n}[1,-1;\alpha ,\gamma ]={S}_{n}(p,\alpha ,\gamma )$ ($n\in {N}_{0}$);

- (6)
${L}_{1,0}^{1,0}[1,-1;\alpha ,\gamma ]=S(\alpha ,\gamma )$ ($0\le \alpha <1$, $\gamma \in {C}^{\ast}$);

- (7)
${L}_{1,0}^{2,1}[1,-1;\alpha ,\gamma ]=K(\alpha ,\gamma )$ ($0\le \alpha <1$, $\gamma \in {C}^{\ast}$);

- (8)
${L}_{1,0}^{1,0}[1,-1;\alpha ,1-\beta ]={S}^{\ast}(\alpha ,\beta )$ ($0\le \alpha <1$, $0\le \beta <1$).

The classes ${S}_{p,q}^{j,l}[A,B;\gamma ]$ and ${U}_{m,n}(\alpha ,A,B)$ were introduced by Goswami and Aouf [4] and Li and Tang [5], respectively. The classes $US(\alpha ,\beta )$ and $UK(\alpha ,\beta )$ were studied recently in [6] (see also [7–12]). The class ${S}_{n}(p,0,\gamma )={S}_{n}(p,\gamma )$ was introduced by Akbulut *et al.* [13]. Also, the classes $S(0,\gamma )=S(\gamma )$ and $K(0,\gamma )=K(\gamma )$ are said to be classes of starlike and convex of complex order $\gamma \ne 0$ in Δ which were considered by Nasr and Aouf [14] and Wiatrowski [15] (see also [16, 17]), and ${S}^{\ast}(0,\beta )={S}^{\ast}(\beta )$ denotes the class of starlike functions of order *β* in Δ.

A majorization problem for the class $S(\gamma )$ has recently been investigated by Altintas *et al.* [18]. Also, majorization problems for the classes ${S}^{\ast}(\beta )$ and ${S}_{p,q}^{j,l}[A,B;\gamma ]$ have been investigated by MacGregor [1] and Goswami and Aouf [4], respectively. Very recently, Goyal and Goswami [19] (see also [20]) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class ${L}_{p,q}^{j,l}[A,B;\alpha ,\gamma ]$.

## 2 Majorization problem for the class ${L}_{p,q}^{j,l}[A,B;\alpha ,\gamma ]$

We begin by proving the following result.

**Theorem 2.1**

*Let the function*$f\in {A}_{p}$

*and suppose that*$g\in {L}_{p,q}^{j,l}[A,B;\alpha ,\gamma ]$.

*If*${D}^{j}{f}^{(q)}(z)$

*is majorized by*${D}^{l}{g}^{(q)}(z)$

*in*Δ,

*and*

*then*

*where*${r}_{0}={r}_{0}(p,q,\alpha ,\gamma ,j,l,A,B)$

*is the smallest positive root of the equation*

*Proof*Suppose that $g\in {L}_{p,q}^{j,l}[A,B;\alpha ,\gamma ]$. Then, making use of the fact that

which holds true for all $z\in \mathrm{\Delta}$.

*P*denotes the well-known class of the bounded analytic functions in Δ and satisfies the conditions

*z*and multiplying by

*z*, we get

*e.g.*, Nehari [21])

Hence, upon setting $\rho =1$ in (2.11), we conclude that (2.1) of Theorem 2.1 holds true for $|z|\le {r}_{0}(p,q,\alpha ,\gamma ,j,l,A,B)$, which completes the proof of Theorem 2.1. □

Setting $\alpha =0$ in Theorem 2.1, we get the following result.

**Corollary 2.1**

*Let the function*$f\in {A}_{p}$

*and suppose that*$g\in {S}_{p,q}^{j,l}[A,B;\gamma ]$.

*If*${D}^{j}{f}^{(q)}(z)$

*is majorized by*${D}^{l}{g}^{(q)}(z)$

*in*Δ,

*and*

*then*

**Remark 2.1** Corollary 2.1 improves the result of Goswami and Aouf [[4], Theorem 1].

Putting $p=1$, $q=0$, $j=m$, $l=n$, $m>n$ and $\gamma =1$ in Theorem 2.1, we obtain the following result.

**Corollary 2.2**

*Let the function*$f\in A$

*and suppose that*$g\in {U}_{m,n}(\alpha ,A,B)$.

*If*${D}^{m}f(z)$

*is majorized by*${D}^{n}g(z)$

*in*Δ,

*then*

For $A=1-2\beta $, $B=-1$, putting $m=1$, $n=0$ and $m=2$, $n=1$ in Corollary 2.2, respectively, we obtain the following Corollaries 2.3 and 2.4.

**Corollary 2.3**

*Let the function*$f\in A$

*and suppose that*$g\in US(\alpha ,\beta )$.

*If*$Df(z)$

*is majorized by*$g(z)$

*in*Δ,

*then*

*where*${r}_{0}={r}_{0}(\alpha ,\beta )$

*is the smallest positive root of the equation*

**Corollary 2.4**

*Let the function*$f\in A$

*and suppose that*$g\in UK(\alpha ,\beta )$.

*If*${D}^{2}f(z)$

*is majorized by*$Dg(z)$

*in*Δ,

*then*

*where*${r}_{0}={r}_{0}(\alpha ,\beta )$

*is the smallest positive root of the equation*

Also, putting $A=1$, $B=-1$, $q=0$, $j=n+1$ and $l=n$ in Theorem 2.1, we obtain the following result.

**Corollary 2.5**

*Let the function*$f\in {A}_{p}$

*and suppose that*$g\in {S}_{n}(p,\alpha ,\gamma )$.

*If*${D}^{n+1}f(z)$

*is majorized by*${D}^{n}g(z)$

*in*Δ,

*then*

## Declarations

### Acknowledgements

Dedicated to Professor Hari M. Srivastava.

The present investigation is partly supported by the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2009MS0113, 2010MS0117. The authors would like to thank the referees for their helpful comments and suggestions to improve our manuscript.

## Authors’ Affiliations

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