- Research
- Open access
- Published:
On some inequalities of a certain class of analytic functions
Journal of Inequalities and Applications volume 2012, Article number: 250 (2012)
Abstract
The aim of this paper is to study the properties of a subclass of analytic functions related to p-valent Bazilevic functions by using the concept of differential subordination. We investigate some results concerned with coefficient bounds, inclusion results, radius problem, covering theorem, angular estimation of a certain integral operator, and some other interesting properties.
MSC:30C45, 30C50.
1 Introduction and preliminaries
Let be the class of analytic functions
defined in the open unit disc . A function is a p-valent starlike function of order ρ if and only if
This class of functions is denoted by . It is noted that . If satisfies
for some and for all , then the function f is called strongly starlike p-valent of order η in E. We denote this class by . Let and be analytic in E. We say is subordinate to , written or , if there exists a Schwarz function , , and in E, then . A function f in is said to belong to the class , , if and only if
For , we obtain the class of Janowski starlike functions. Janowski functions have extensively been studied by several researchers; see for example [1–3]. It is clear that if and only if
and
A function is p-valent Bazilevic function of type and order ρ if and only if
where , and . For and , this class was introduced by Bazilevic and these functions are univalent for , . This class of functions is studied by many authors; for some details, see [4–11].
Using this concept, we generalize and define a subclass of p-valent Bazilevic functions of type as follows.
Definition 1.1 A function if it satisfies the condition
where , , , and β is any real.
We have the following special cases.
-
(i)
For , we have the subclass of Bazilevic functions defined by Patel [12].
-
(ii)
For , , , , , we obtain the subclass of Bazilevic functions defined in [13].
For , , we have the following subclass of analytic functions.
Definition 1.2 A function if it satisfies the condition
where , , , , β is any real and . In other words, a function if it satisfies the condition
We need the following definition and lemmas which will be used in our main results.
Definition 1.3 Let be analytic in a domain D and h be univalent in E. If p is analytic in E with when , then we say that p satisfies a first-order differential subordination if
The univalent function q is called dominant of the differential subordination (1.4) if for all p satisfies (1.4). If for all dominants of (1.4), then we say that is the best dominant of (1.4).
Lemma 1.4 ([14])
If , and the complex number γ satisfies , then the differential equation
has a univalent solution in E given by
If is analytic in E and satisfies
then
and is the best dominant.
Lemma 1.5 ([15])
Let ε be a positive measure on . Let g be a complex-valued function defined on such that is analytic in E for each and is ε-integrable on for all . In addition, suppose that , is real and for and . If , then .
Lemma 1.6 ([[16], Chapter 14])
Let , and be complex numbers. Then, for ,
Lemma 1.7 ([17])
Let . Then
Lemma 1.8 ([18])
Let F be analytic and convex in E. If and , then
Lemma 1.9 ([19])
Let be analytic in E and be analytic and convex in E. If , then
Lemma 1.10 ([20])
Let be analytic in E and in E. If there exists a point such that () and (), then we have , where
and ().
Lemma 1.11 Let . Then the function
belongs to for .
Proof is straightforward by using Lemma 1.4.
Throughout this paper, , , , and unless otherwise stated.
2 Main results
Theorem 2.1 If , then
where and
In hypergeometric function form,
and if , , then , where
This result is best possible.
Proof Let
where is analytic in E with . Differentiating logarithmically, we obtain
Using Lemma 1.4 for and , we have
where is given in (2.3) and is the best dominant of (2.5). Next, in order to prove , we show that . Now, we set , and , then it is clear that ; therefore, for it follows from (2.2) by using Lemma 1.6 that
To prove that , we need to show that
Since with implies that , therefore, by using Lemma 1.6, (2.6) yields
where
which is a positive measure on . For it is clear that and is real for and . Also,
for . Therefore, using Lemma 1.5, we have
Now, letting , it follows
Therefore, . □
For , we have the following result proved in [12].
Corollary 2.2 If , then
and if , , then , where
This result is best possible.
For , we have the class . We denote the class of functions , having Taylor series representation of the form
and satisfying the condition
by , where such that . Now, we derive the following result for the class .
Theorem 2.3 Let . Then
Proof Since , therefore,
Now, using the fact that and , we obtain
By a well-known result due to Janowski and Lemma 1.9, we have
By the triangle inequality, we obtain
Using the coefficient bound for the class , we have the required result. □
For and , we have the following result proved in [13].
Corollary 2.4 Let . Then
For , , , and , we have the following result proved in [21].
Corollary 2.5 Let f satisfy the condition
Then
Theorem 2.6 For and ,
Proof Let . Then
Since , therefore by Lemma 1.7, we have
Hence, we have . For , we have the required result. When , Theorem 2.1 implies that
Now
Using Lemma 1.8, we have the required result. □
For , we have the following result.
Corollary 2.7 For and ,
This result is proved in [13].
For , , , and , we have the class defined as
for . Now have the following result for the class proved in [21].
Corollary 2.8 For , and ,
Theorem 2.9 Let satisfy
for . Then f is p-valent convex in , where
Proof Let
where is analytic in E with and . By using the Schwarz lemma, we get
where is analytic in E with . Differentiating logarithmically, we have
Since , therefore,
This implies that
Now, using the well-known results for classes , P and the Schwarz function [22], we have
Let . Since and , therefore, and . It follows that the root lies in . This implies that if , where is given by (2.10). □
For and , we have the following result which is proved in [12].
Corollary 2.10 Let satisfy
. Then f is p-valent convex in , where
Theorem 2.11 Let satisfy
for . Then, for , f is p-valent -convex in , where
Proof Let
where is analytic in E with and . By using the Schwarz lemma, we get
where is analytic in E with . Differentiating logarithmically, we have
This implies that
Now, using the well-known results for classes , and the Schwarz function, we have
Let . Then for , and , and . It shows that the root lies in . This implies that if , where is given by (2.11). □
Theorem 2.12 Let . Then E is mapped by f on a domain that contains the disc , where
Proof Let be any complex number such that . Then
is univalent in E, so that
Therefore,
Hence,
□
For , , , , we have the following result proved in [13].
Corollary 2.13 Let . Then E is mapped by f on a domain that contains the disc , where
Theorem 2.14 Let , and let . If
for some , then
where
with
and
Proof Since
therefore,
Now using (1.5), we have
Let
where is analytic with . Now
where
Since , therefore, we can write
where
similarly
Suppose that in E, there exists a point such that () and . Now, using Lemma 1.10, we have . At first suppose that () for the case , we obtain
where υ and are given by (2.16) and (2.17) respectively. For , we have
which is a contradiction to the assumption of our theorem. Now we suppose that . For the case using a similar method, we obtain
and for , we have
which is a contradiction to the assumption of our theorem. Hence, we have the proof. □
This kind of problem is also considered in [23]. For , we have the following result proved in [24].
Corollary 2.15 Let , and let . If
then
where
with
and
Remark 2.16 By using the suitable choices of parameters c, α, A and B, we can find many results proved in the literature.
References
Çag̃lar M, Polatog̃lu Y, Şen A, Yavuz E, Owa S: On Janowski starlike functions. J. Inequal. Appl. 2007. doi:10.1155/2007/14630
Sokół J: On a class of analytic multivalent functions. Appl. Math. Comput. 2008, 203: 210–216. 10.1016/j.amc.2008.04.027
Sokół J: Classes of multivalent functions associated with a convolution operator. Comput. Math. Appl. 2010, 60: 1343–1350. 10.1016/j.camwa.2010.06.015
Arif M, Noor KI, Raza M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl. 2012. doi:10.1186/1029–242X-2012–22
Arif M, Noor KI, Raza M: On a class of analytic functions related with generalized Bazilevic type functions. Comput. Math. Appl. 2011, 61: 2456–2462. 10.1016/j.camwa.2011.02.026
Arif M, Raza M, Noor KI, Malik SN: On strongly Bazilevic functions associated with generalized Robertson functions. Math. Comput. Model. 2011, 54: 1608–1612. 10.1016/j.mcm.2011.04.033
Bazilevic IE: On a class of integrability in quadratures of the Loewner-Kufarev equation. Mat. Sb. N.S. 1955, 37: 471–476.
El-Ashwah RM, Aouf MK: Some properties of certain classes of meromorphically p -valent functions involving extended multiplier transformations. Comput. Math. Appl. 2010, 59: 2111–2120. 10.1016/j.camwa.2009.12.016
Noor KI: On Bazilevic functions of complex order. Nihonkai Math. J. 1992, 3: 115–124.
Noor KI: Bazilevic functions of type β . Int. J. Math. Math. Sci. 1982, 5(2):411–415. 10.1155/S0161171282000416
Raza M, Noor KI: A class of Bazilevic type functions defined by convolution operator. J. Math. Inequal. 2011, 5(2):253–261.
Patel J: On certain subclass of p -valently Bazilevic functions. J. Inequal. Pure Appl. Math. 2005., 6(1): Article ID 16
Wang Z-G, Jiang Y-P: Notes on certain subclass of p -valently Bazilevic functions. J. Inequal. Pure Appl. Math. 2008., 9(3): Article ID 70
Miller SS, Mocanu PT: Univalent solutions of Briot-Bouquet differential subordination. J. Differ. Equ. 1985, 56: 297–309. 10.1016/0022-0396(85)90082-8
Wilken DR, Feng J: A remark on convex and starlike functions. J. Lond. Math. Soc. 1980, 21: 287–290. 10.1112/jlms/s2-21.2.287
Whittaker ET, Watson GN: A Course of Modern Analysis. 4th edition. Cambridge University Press, Cambridge; 1958.
Liu M-S: On a subclass of p -valent close-to-convex functions of order β and type α . J. Math. Study 1997, 30: 102–104. in Chinese
Liu M-S: On certain subclass of analytic functions. J. South China Norm. Univ. 2002, 4: 15–20. in Chinese
Rogosinski W: On coefficient of subordinate functions. Proc. Lond. Math. Soc. 1943, 48: 48–82.
Nunokawa M: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 1993, 68: 234–237.
Guo D, Liu M-S: On certain subclass of Bazilevic functions. J. Inequal. Pure Appl. Math. 2007., 8(1): Article ID 12
Causey WM, Merkes EP: Radii of starlikeness for certain classes of analytic functions. J. Math. Anal. Appl. 1970, 31: 579–586. 10.1016/0022-247X(70)90010-7
El-Ashwah RM: Results regarding the argument of certain p -valent analytic functions defined by a generalized integral operator. J. Inequal. Appl. 2012., 2012: Article ID 35. doi:10.1186/1029–242X-2012–35
Cho NE, Kim IH, Kim JA: Angular estimation of certain integral operators. Int. J. Math. Math. Sci. 1998, 21(2):369–374. 10.1155/S0161171298000507
Acknowledgements
The third author would like to express gratitude to Dr. S.M. Junaid Zaidi, Rector CIIT for his support and for providing excellent research facilities and to the Higher Education Commission of Pakistan for financial assistance.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MR, SNM and KIN jointly discussed and presented the ideas of this article. MR made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Raza, M., Malik, S.N. & Noor, K.I. On some inequalities of a certain class of analytic functions. J Inequal Appl 2012, 250 (2012). https://doi.org/10.1186/1029-242X-2012-250
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-250