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Majorization properties for certain new classes of analytic functions using the Salagean operator
Journal of Inequalities and Applications volume 2013, Article number: 86 (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.
1 Introduction and definitions
Let f and g be two analytic functions in the open unit disk
We say that f is majorized by g in Δ (see [1]) and write
if there exists a function φ, analytic in Δ, such that
It may be noted here that (1.2) is closely related to the concept of quasisubordination between analytic functions.
For two 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
such that
We denote this subordination by f(z)\prec g(z). Furthermore, if the function g is univalent in Δ, then
Let {A}_{p} denote the class of functions of the form
that are analytic and pvalent in the open unit disk Δ. Also, let {A}_{1}=A.
For a function f\in {A}_{p}, let {f}^{(q)} denote a q thorder ordinary differential operator by
where p>q, p\in N, q\in {N}_{0}=N\cup \{0\} and z\in \mathrm{\Delta}. Next, Frasin [2] introduced the differential operator {D}^{m}{f}^{(q)} as follows:
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 pvalent functions of complex order \gamma \ne 0 in Δ if and only if
Clearly, we have the following relationships:

(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}[12\beta ,1;\alpha ,1]=US(\alpha ,\beta ) (0\le \beta <1) (αuniformly starlike functions of order β);

(4)
{L}_{2,1}^{1,0}[12\beta ,1;\alpha ,1]=UK(\alpha ,\beta ) (0\le \beta <1) (αuniformly 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
and letting
in (1.7), we obtain
or, equivalently,
which holds true for all z\in \mathrm{\Delta}.
We find from (2.3) that
where \omega (z)={c}_{1}z+{c}_{2}{z}^{2}+\cdots , \omega \in P, P denotes the wellknown class of the bounded analytic functions in Δ and satisfies the conditions
From (2.4), we get
By virtue of (2.5), we obtain
Next, since {D}^{j}{f}^{(q)}(z) is majorized by {D}^{l}{g}^{(q)}(z) in Δ, thus from (1.3), we have
Differentiating the above equality with respect to z and multiplying by z, we get
Thus, by noting that \phi (z)\in P satisfies the inequality (see, e.g., Nehari [21])
and making use of (2.6) and (2.8) in (2.7), we obtain
which, upon setting
leads us to the inequality
where
takes its maximum value at \rho =1 with {r}_{0}={r}_{0}(p,q,\alpha ,\gamma ,j,l,A,B), where
is the smallest positive root of equation (2.2). Furthermore, if 0\le \delta \le {r}_{0}(p,q,\alpha ,\gamma ,j,l,A,B), then the function \psi (\rho ) defined by
is an increasing function on the interval 0\le \rho \le 1 so that
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
where {r}_{0}={r}_{0}(p,q,\gamma ,j,l,A,B) is the smallest positive root of the equation
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
where {r}_{0}={r}_{0}(\alpha ,A,B) is the smallest positive root of the equation
For A=12\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
where {r}_{0}={r}_{0}(p,\alpha ,\gamma ) is the smallest positive root of the equation
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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.
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Li, SH., Tang, H. & Ao, E. Majorization properties for certain new classes of analytic functions using the Salagean operator. J Inequal Appl 2013, 86 (2013). https://doi.org/10.1186/1029242X201386
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DOI: https://doi.org/10.1186/1029242X201386
Keywords
 analytic functions
 multivalent functions
 αuniformly starlike functions of order β
 αuniformly convex functions of order β
 subordination
 majorization property