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On a new subclass of Ruscheweyh-type harmonic multivalent functions
Journal of Inequalities and Applications volume 2013, Article number: 271 (2013)
Abstract
We introduce a certain subclass of harmonic multivalent functions defined by using a Ruscheweyh derivative operator. We obtain coefficient conditions, distortion bounds, extreme points, convex combination for the above class of harmonic multivalent functions. We also derive inclusion relationships involving the neighborhoods of harmonic multivalent functions belonging to this subclass.
MSC:30C45, 30C50.
Dedication
Dedicated to Professor Hari M Srivastava
1 Introduction
A continuous function is a complex-valued harmonic function in a domain if both u and v are real harmonic in D. In any simply connected domain D, we can write , where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. The harmonic function is sense preserving and locally one-to-one in D if in D. See Clunie and Sheil-Small [1].
For , , denote by the class of functions that are sense preserving, harmonic multivalent in the unit disk , where h and g are defined by
which are analytic and multivalent functions in U.
Also, denote by the subclass of consisting of harmonic functions , where h and g are of the form
Note that reduces to , the class of analytic multivalent functions with negative coefficients, if the co-analytic part of is identically zero.
We define an extended linear derivative operator of a Ruscheweyh-type harmonic function in by
where D is the Ruscheweyh derivative [2] of power series , given by
The operator ∗ stands for the Hadamard product or convolution of two power series
defined by
Raina and Srivastava [3] introduced this extended Ruscheweyh operator for the class .
Next, we define the ordinary differential operator to be
where
Let denote the subclass of consisting of functions that satisfy the condition
Define .
Taking the co-analytic part of identically zero and specializing the parameters, we obtain the following subclasses:
We will use the notations
Following Goodman [7] and Ruscheweyh [8] (see also [9–11] and [12]), for , we define the set of the δ-neighborhood of ,
In particular, for the function , we immediately have
Ruscheweyh-type harmonic univalent functions have been studied by several authors such as [13, 14] and [15]. The object of the present paper is to investigate the various properties of multivalent harmonic functions belonging to the subclass . This class is motivated by two earlier investigations [5] and [3]. We extend the results of [5] which include harmonic multivalent functions. Necessary and sufficient coefficient conditions, distortion bounds, extreme points and convex combination of the above mentioned class are derived. Also, inclusion relationships involving the neighborhoods of multivalent harmonic functions belonging to this subclass are established.
2 Main results
Denote by the class of functions of the form (1) which are sense preserving and multivalent harmonic starlike, satisfying the condition , for each , , and .
Lemma 2.1 Let be of the form (1). If
then .
Remark Lemma 2.1 follows immediately from the result due to Ahuja [16] upon setting p and k instead of m and , respectively.
Theorem 2.2 Let be given by (1). Furthermore, let
then f is sense preserving, harmonic multivalent in U, and , where
Proof If the inequality (9) holds for the coefficients of , then by (8), f is sense preserving, harmonic multivalent and starlike in U. In view of (5), we need to prove that , where
Using the fact that , it suffices to show that
Therefore we obtain
This last expression is non-negative by (9), and so the proof is complete. □
Corollary 2.3 For and , if the inequality
holds, then f is sense preserving, harmonic multivalent in U, and .
Corollary 2.4 For , if the inequality
holds, then f is sense preserving, harmonic multivalent in U, and .
Theorem 2.5 Let be given by (2). Then
-
(i)
for and , if and only if
(10) -
(ii)
for , if and only if
(11)
Proof The ‘if’ part follows from Theorem 2.2, Corollary 2.3 and Corollary 2.4 upon noting that . For the ‘only if’ part, we show that if the condition (11) does not hold.
Note that a necessary and sufficient condition for given by (2) to be in is that the condition (5) to be satisfied. This is equivalent to
The above condition must hold for all values of z, . Upon choosing the values of z on the positive real axis where , we must have
If the condition (11) does not hold, then the numerator of (12) is negative for r sufficiently close to 1 because of conditions (i) or (ii). Thus there exists a in , for which the quotient in (12) is negative. This contradicts the required condition for and so the proof is complete. □
Next we determine the distortion bounds for the functions in .
Theorem 2.6 Let . Then for , we have
-
(i)
for and ,
and
-
(ii)
for ,
and
Proof (i) We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let . Taking the absolute value of f, we have
Using Theorem 2.5(i), we obtain
The proof of the other case is similar and so is omitted. □
The following covering result follows from the left-hand inequality in Theorem 2.6.
Corollary 2.7 Let f of the form (2) be so that . Then
-
(i)
for and ,
-
(ii)
for ,
Theorem 2.8 Let f be given by (2). Then if and only if
where , , for is of the form
and , for is of the form
In particular, the extreme points of are and .
Proof Suppose , , and
Then
and so .
Conversely, if , then
and
Set
and
where . Then, as required, we obtain
The proof for the case is similar and hence is omitted. □
Theorem 2.9 The class is closed under convex combinations.
Proof Let for , where is given by
Then by (10) and (11),
For , , the convex combination of may be written as
Then by (13),
This is the condition required by (10) and (11), and so . □
Theorem 2.10 Let , then
where and are given by (7),
-
(i)
for and ,
-
(ii)
for ,
where
Proof Let , and . We need to show that . It suffices to show that s satisfies the condition (7). In view of Theorem 2.5(i), we have
Then
so that
which, in view of definition (7), completes the proof of Theorem 2.10. The proof of other case is similar and so is omitted. □
Remark Taking the co-analytic part of of the form (7) identically zero and letting , we obtain the neighborhood result of Ashwah [5].
Remark Taking the co-analytic part of of the form (7) identically zero and letting , we obtain the neighborhood result of Murugusundaramoorthy and Srivastava [6].
Theorem 2.11 Let and
-
(i)
for and ,
-
(ii)
for ,
then
where
Proof Let and . Also, suppose that and . We need to show that . It suffices to show that s satisfies the condition (8). We have
Now this expression is never greater than p provided that
The proof of the other case is similar and so is omitted. □
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Yaşar, E., Yalçın, S. On a new subclass of Ruscheweyh-type harmonic multivalent functions. J Inequal Appl 2013, 271 (2013). https://doi.org/10.1186/1029-242X-2013-271
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DOI: https://doi.org/10.1186/1029-242X-2013-271