A class of analytic functions involving in the Dziok-Srivastava operator
© Xu et al.; licensee Springer 2013
Received: 7 December 2012
Accepted: 8 March 2013
Published: 1 April 2013
Let be a class of functions of the form
which are analytic in the open unit disk . By means of the Dziok-Srivastava operator, we introduce a new subclass
of . In particular, coincides with the class of uniformly convex functions introduced by Goodman. The order of starlikeness and the radius of α-spirallikeness of order β () are computed. Inclusion relations and convolution properties for the class are obtained. A special member of is also given. The results presented here not only generalize the corresponding known results, but also give rise to several other new results.
Keywordsuniformly convex functions convex univalent functions starlike functions α-spirallike functions convolution Dziok-Srivastava operator subordination
In this paper we introduce and investigate the following subclass of .
Throughout this paper we assume, unless otherwise stated, that l, m, α and μ satisfy (1.10).
2 Subordination theorem
It follows from (2.2) that . In order to prove the theorem, it suffices to show that the function given by (2.2) maps conformally onto the parabolic region Ω.
maps conformally onto Ω. The proof of the theorem is now completed. □
The results are sharp.
where is analytic and in .
for . Hence we have (2.7) and (2.8).
shows that the estimates (2.7) and (2.8) are sharp. □
where is analytic in with and ().
from (2.13) and (1.7), we obtain (2.12). □
3 Properties of the class
and the order is sharp.
and the order in (3.1) is sharp for the function defined by (2.11). □
The result is sharp.
that is, is α-spirallike of order β in . Also, the result is sharp for the function defined by (2.11). □
Setting , Theorem 3 reduces to the following.
Corollary 3 Let . Then is α-spirallike of order in . The result is sharp.
where denotes the convex hull of .
Applying the lemma, we derive Theorems 4 and 5 below.
Therefore, by Theorem 1, and (3.8) is proved. □
Consequently, by applying Theorem 1, and the proof of (3.13) is completed. □
Thus Theorem 5 yields the following.
- (i)If and , then
- (ii)If with and , then
The result is sharp, that is, cannot be increased.
Now, by virtue of (3.19), (3.20), (3.21), and (i)-(iv), we have proved the theorem. □
The result is sharp.
is analytic in , and , then (). Hence (3.33) follows from (3.36) at once. □
The estimate (3.33) is sharp since equality is attained for the function defined by (2.11).
Dedicated to Professor Hari M Srivastava.
This work was partially supported by the National Natural Science Foundation of China (Grant No. 11171045).
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