International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 369078 | 15 pages | https://doi.org/10.1155/2010/369078

Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions

Academic Editor: Dorothy Wallace
Received09 Jun 2009
Revised20 Dec 2009
Accepted09 Mar 2010
Published07 Apr 2010

Abstract

A new generalised derivative operator šœ‡šœ†š‘›,š‘š1,šœ†2 is introduced. This operator generalised many well-known operators studied earlier by many authors. Using the technique of differential subordination, we will study some of the properties of differential subordination. In addition we investigate several interesting properties of the new generalised derivative operator.

1. Introduction and Preliminaries

Let š’œ denote the class of functions of the form

š‘“(š‘§)=š‘§+āˆžī“š‘˜=2š‘Žš‘˜š‘§š‘˜,whereš‘Žš‘˜iscomplexnumber,(1.1) which are analytic in the open unit disc š‘ˆ={š‘§āˆˆā„‚āˆ¶|š‘§|<1} on the complex plane ā„‚. Let š‘†,š‘†āˆ—(š›¼),š¶(š›¼)(0ā‰¤š›¼<1) denote the subclasses of š’œ consisting of functions that are univalent, starlike of order š›¼, and convex of order š›¼ in š‘ˆ, respectively. In particular, the classes š‘†āˆ—(0)=š‘†āˆ—andš¶(0)=š¶ are the familiar classes of starlike and convex functions in š‘ˆ, respectively. A function š‘“āˆˆš¶(š›¼) if Re(1+š‘§š‘“ī…žī…ž/š‘“ī…ž)>š›¼. Furthermore a function š‘“ analytic in š‘ˆ is said to be convex if it is univalent and š‘“(š‘ˆ) is convex.

Let ā„‹(š‘ˆ) be the class of holomorphic function in unit disc š‘ˆ={š‘§āˆˆā„‚āˆ¶|š‘§|<1}.

Let

š’œš‘›=ī€½š‘“āˆˆā„‹(š‘ˆ)āˆ¶š‘“(š‘§)=š‘§+š‘Žš‘›+1š‘§š‘›+1ī€¾,+ā‹Æ,(š‘§āˆˆš‘ˆ)(1.2) with š’œ1=š’œ.

For š‘Žāˆˆā„‚ and š‘›āˆˆā„•={1,2,3,ā€¦} we let

ā„‹[]=ī€½š‘Ž,š‘›š‘“āˆˆā„‹(š‘ˆ)āˆ¶š‘“(š‘§)=š‘§+š‘Žš‘›š‘§š‘›+š‘Žš‘›+1š‘§š‘›+1ī€¾.+ā‹Æ,(š‘§āˆˆš‘ˆ)(1.3) Let āˆ‘š‘“(š‘§)=š‘§+āˆžš‘˜=2š‘Žš‘˜š‘§š‘˜āˆ‘andš‘”(š‘§)=š‘§+āˆžš‘˜=2š‘š‘˜š‘§š‘˜be analytic in the open unit disc š‘ˆ={š‘§āˆˆā„‚āˆ¶|š‘§|<1}. Then the Hadamard product (or convolution) š‘“āˆ—š‘” of the two functions š‘“, š‘” is defined by

š‘“(š‘§)āˆ—š‘”(š‘§)=(š‘“āˆ—š‘”)(š‘§)=š‘§+āˆžī“š‘˜=2š‘Žš‘˜š‘š‘˜š‘§š‘˜.(1.4) Next, we state basic ideas on subordination. If š‘“ and š‘” are analytic in š‘ˆ, then the function š‘“ is said to be subordinate to š‘”, and can be written as

š‘“ā‰ŗš‘”,š‘“(š‘§)ā‰ŗš‘”(š‘§),(š‘§āˆˆš‘ˆ),(1.5) if and only if there exists the Schwarz function š‘¤, analytic in š‘ˆ, with š‘¤(0)=0and|š‘¤(š‘§)|<1such thatš‘“(š‘§)=š‘”(š‘¤(š‘§)), (š‘§āˆˆš‘ˆ).

Furthermore if š‘” is univalent in š‘ˆ, then š‘“ā‰ŗš‘” if and only if š‘“(0)=š‘”(0) and š‘“(š‘ˆ)āŠ‚š‘”(š‘ˆ) (see [1, page 36]).

Let šœ“āˆ¶ā„‚3Ɨš‘ˆā†’ā„‚ and let ā„Ž be univalent in š‘ˆ. If š‘ is analytic in š‘ˆ and satisfies the (second-order) differential subordination

šœ“ī€·š‘(š‘§),š‘§š‘ī…ž(š‘§),š‘§2š‘ī…žī…žī€ø(š‘§);š‘§ā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(1.6) then š‘ is called a solution of the differential subordination.

The univalent function š‘ž is called a dominant of the solutions of the differential subordination, or more simply a dominant, if š‘ā‰ŗš‘ž for all š‘ satisfying (1.6). A dominant Ģƒš‘ž that satisfies Ģƒš‘žā‰ŗš‘ž for all dominants š‘ž of (1.6) is said to be the best dominant of (1.6). (Note that the best dominant is unique up to a rotation of š‘ˆ.)

Now, (š‘„)š‘˜ denotes the Pochhammer symbol (or the shifted factorial) defined by

(š‘„)š‘˜=īƒÆ1forš‘˜=0,š‘„āˆˆā„‚ā§µ{0},š‘„(š‘„+1)(š‘„+2)ā‹Æ(š‘„+š‘˜āˆ’1)forš‘˜āˆˆā„•={1,2,3,ā€¦},š‘„āˆˆā„‚.(1.7) To prove our results, we need the following equation throughout the paper:

šœ‡šœ†š‘›,š‘š+11,šœ†2ī€·š‘“(š‘§)=1āˆ’šœ†1ī€øī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)+šœ†1š‘§ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž,(š‘§āˆˆš‘ˆ),(1.8) where š‘›,š‘šāˆˆā„•0={0,1,2,ā€¦}, šœ†2ā‰„šœ†1ā‰„0, and šœ™šœ†2(š‘§) is analytic function given by

šœ™šœ†2(š‘§)=š‘§+āˆžī“š‘˜=2š‘§š‘˜1+šœ†2.(š‘˜āˆ’1)(1.9) Here šœ‡šœ†š‘›,š‘š1,šœ†2 is the generalized derivative operator which we shall introduce later in the paper. Moreover, we need the following lemmas in proving our main results.

Lemma 1.1 (see [2, page 71]). Let ā„Ž be analytic, univalent, and convex in š‘ˆ, with ā„Ž(0)=š‘Ž, š›¾ā‰ 0 and, Reš›¾ā‰„0. If š‘āˆˆā„‹[š‘Ž,š‘›]and š‘(š‘§)+š‘§š‘ī…ž(š‘§)š›¾ā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(1.10) then š‘(š‘§)ā‰ŗš‘ž(š‘§)ā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(1.11) where š‘ž(š‘§)=(š›¾/š‘›š‘§š›¾/š‘›)āˆ«š‘§0ā„Ž(š‘”)š‘”(š›¾/š‘›)āˆ’1š‘‘š‘”,(š‘§āˆˆš‘ˆ).

The function š‘ž is convex and is the best (š‘Ž,š‘›)-dominant.

Lemma 1.2 (see [3]). Let š‘” be a convex function in š‘ˆ and let ā„Ž(š‘§)=š‘”(š‘§)+š‘›š›¼š‘§š‘”ā€²(š‘§),(1.12) where š›¼>0 and š‘› is a positive integer.
If
š‘(š‘§)=š‘”(0)+š‘š‘›š‘§š‘›+š‘š‘›+1š‘§š‘›+1+ā‹Æ,(š‘§āˆˆš‘ˆ),(1.13) is holomorphic in š‘ˆ and š‘(š‘§)+š›¼š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(1.14) then š‘(š‘§)ā‰ŗš‘”(š‘§),(1.15) and this result is sharp.

Lemma 1.3 (see [4]). Let š‘“āˆˆš’œ, if ī‚µRe1+š‘§š‘“ī…žī…ž(š‘§)š‘“ī…žī‚¶1(š‘§)>āˆ’2,(1.16) then 2š‘§ī€œš‘§0š‘“(š‘”)š‘‘š‘”,(š‘§āˆˆš‘ˆ,š‘§ā‰ 0),(1.17) belongs to the class of convex functions.

2. Main Results

In the present paper, we will use the method of differential subordination to derive certain properties of generalised derivative operator šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§). Note that differential subordination has been studied by various authors, and here we follow similar works done by Oros [5] and G. Oros and G. I. Oros [6].

In order to derive our new generalised derivative operator, we define the analytic function

š¹š‘ššœ†1,šœ†2(š‘§)=š‘§+āˆžī“š‘˜=2ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘šī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šāˆ’1š‘§š‘˜,(2.1) where š‘šāˆˆā„•0={0,1,2,ā€¦} and šœ†2ā‰„šœ†1ā‰„0. Now, we introduce the new generalised derivative operator šœ‡šœ†š‘›,š‘š1,šœ†2 as follows.

Definition 2.1. For š‘“āˆˆš’œ the operator šœ‡šœ†š‘›,š‘š1,šœ†2 is defined by šœ‡šœ†š‘›,š‘š1,šœ†2āˆ¶š’œā†’š’œšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)=š¹š‘ššœ†1,šœ†2(š‘§)āˆ—š‘…š‘›š‘“(š‘§),(š‘§āˆˆš‘ˆ),(2.2) where š‘›,š‘šāˆˆā„•0=ā„•āˆŖ{0},šœ†2ā‰„šœ†1ā‰„0,andš‘…š‘›š‘“(š‘§) denotes the Ruscheweyh derivative operator [7], given by š‘…š‘›š‘“(š‘§)=š‘§+āˆžī“š‘˜=2š‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜,ī€·š‘›āˆˆā„•0ī€ø,,š‘§āˆˆš‘ˆ(2.3) where š‘(š‘›,š‘˜)=(š‘›+1)š‘˜āˆ’1/(1)š‘˜āˆ’1.
If š‘“ is given by (1.1), then we easily find from equality (2.2) that
šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)=š‘§+āˆžī“š‘˜=2ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘šī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šāˆ’1š‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜,(š‘§āˆˆš‘ˆ),(2.4) where š‘›,š‘šāˆˆā„•0={0,1,2ā€¦},šœ†2ā‰„šœ†1ā‰„0,andš‘(š‘›,š‘˜)=(š‘›+š‘˜āˆ’1š‘›)=(š‘›+1)š‘˜āˆ’1/(1)š‘˜āˆ’1.

Remark 2.2. Special cases of this operator include the Ruscheweyh derivative operator in two cases šœ‡š‘›,10,šœ†2ā‰”š‘…š‘› and šœ‡šœ†š‘›,01,0ā‰”š‘…š‘› [7], the Salagean derivative operator šœ‡0,š‘š1,0ā‰”š‘†š‘› [8], the generalised Ruscheweyh derivative operator in two cases šœ‡šœ†š‘›,11,šœ†2ā‰”š‘…š‘›šœ† and šœ‡šœ†š‘›,01,šœ†2ā‰”š‘…š‘›šœ† [9], the generalised Salagean derivative operator šœ‡šœ†0,š‘š1,0ā‰”š‘†š‘›š›½ introduced by Al-Oboudi [10], and the generalised Al-Shaqsi and Darus derivative operator šœ‡šœ†š‘›,š‘š1,0ā‰”š·š‘›šœ†,š›½ that can be found in [11].

Now, we remind the well-known Carlson-Shaffer operator šæ(š‘Ž,š‘) [12] associated with the incomplete beta function šœ™(š‘Ž,š‘;š‘§), defined by

šæšæ(š‘Ž,š‘)āˆ¶š’œā†’š’œ,(š‘Ž,š‘)š‘“(š‘§)āˆ¶=šœ™(š‘Ž,š‘;š‘§)āˆ—š‘“(š‘§),(š‘§āˆˆš‘ˆ),(2.5) where

šœ™(š‘Ž,š‘;š‘§)=š‘§+āˆžī“š‘˜=2(š‘Ž)š‘˜āˆ’1(š‘)š‘˜āˆ’1š‘§š‘˜,(2.6)š‘Ž is any real number, and š‘āˆ‰š‘§āˆ’0;š‘§āˆ’0={0,āˆ’1,āˆ’2,ā€¦}.

It is easily seen that

šœ‡šœ†0,01,0š‘“(š‘§)=šœ‡0,š‘š0,0š‘“(š‘§)=šœ‡0,10,šœ†2š‘“(š‘§)=šœ‡1,20,1šœ‡š‘“(z)=šæ(š‘Ž,š‘Ž)š‘“(š‘§)=š‘“(š‘§),šœ†1,01,0š‘“(š‘§)=šœ‡1,š‘š0,0š‘“(š‘§)=šœ‡1,10,šœ†2š‘“(š‘§)=šœ‡šœ†0,01,1š‘“(š‘§)=šæ(2,1)š‘“(š‘§)=š‘§š‘“ī…ž(š‘§),(2.7) and also

šœ‡šœ†š‘Žāˆ’1,01,0š‘“(š‘§)=šœ‡š‘Žāˆ’1,10,šœ†2š‘“(š‘§)=šœ‡š‘Žāˆ’1,š‘š0,0š‘“(š‘§)=šæ(š‘Ž,1)š‘“(š‘§),(2.8) where š‘Ž=1,2,3,ā€¦.

Next, we give the following.

Definition 2.3. For š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0, and 0ā‰¤š›¼<1, let š‘…šœ†š‘›,š‘š1,šœ†2(š›¼) denote the class of functions š‘“āˆˆš’œ which satisfy the condition ī‚€šœ‡Rešœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…ž>š›¼,(š‘§āˆˆš‘ˆ).(2.9) Also let š¾šœ†š‘›,š‘š1,šœ†2(š›æ) denote the class of functions š‘“āˆˆš’œ which satisfy the condition ī‚€šœ‡Rešœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚(š‘§)ī…ž>š›æ,(š‘§āˆˆš‘ˆ).(2.10)

Remark 2.4. It is clear that š‘…šœ†0,11,0(š›¼)ā‰”š‘…(šœ†1,š›¼), and the class of functions š‘“āˆˆš’œ satisfying ī€·šœ†Re1š‘§š‘“ī…žī…ž(š‘§)+š‘“ī…žī€ø(š‘§)>š›¼,(š‘§āˆˆš‘ˆ)(2.11) is studied by Ponnusamy [13] and others.

Now we begin with the first result as follows.

Theorem 2.5. Let ā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§1+š‘§,(š‘§āˆˆš‘ˆ),(2.12) be convex in š‘ˆ, with h(0)=1 and 0ā‰¤š›¼<1. If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0, and the differential subordination ī‚€šœ‡š‘›,šœ†š‘š+11,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(2.13) holds, then ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚(š‘§)ī…žā‰ŗš‘ž(š‘§)=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1š‘§1/šœ†1šœŽī‚µ1šœ†1ī‚¶,(2.14) where šœŽ is given by ī€œšœŽ(š‘„)=š‘§0š‘”š‘„āˆ’11+š‘”š‘‘š‘”,(š‘§āˆˆš‘ˆ).(2.15) The function š‘ž is convex and is the best dominant.

Proof. By differentiating (1.8), with respect to š‘§, we obtain ī‚€šœ‡šœ†š‘›,š‘š+11,šœ†2ī‚š‘“(š‘§)ī…ž=ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž+šœ†1š‘§ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žī…ž.(2.16) Using (2.16) in (2.13), differential subordination (2.13) becomes ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž+šœ†1š‘§ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žī…žā‰ŗā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§1+š‘§.(2.17) Let ī‚ƒšœ‡š‘(š‘§)=šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž=īƒ¬š‘§+āˆžī“š‘˜=2ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘šī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šš‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜īƒ­ī…ž=1+š‘1š‘§+š‘2š‘§2[]+ā‹Æ,(š‘āˆˆā„‹1,1,š‘§āˆˆš‘ˆ).(2.18) Using (2.18) in (2.17), the differential subordination becomes š‘(š‘§)+šœ†1š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§.1+š‘§(2.19) By using Lemma 1.1, we have 1š‘(š‘§)ā‰ŗš‘ž(š‘§)=šœ†1š‘§1/šœ†1ī€œš‘§0ā„Ž(š‘”)š‘”(1/šœ†1)āˆ’1=1š‘‘š‘”šœ†1š‘§1/šœ†1ī€œš‘§0ī‚µ1+(2š›¼āˆ’1)š‘”ī‚¶š‘”1+š‘”(1/šœ†1)āˆ’1š‘‘š‘”=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1š‘§1/šœ†1šœŽī‚µ1šœ†1ī‚¶,(2.20) where šœŽ is given by (2.15), so we get ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žā‰ŗš‘ž(š‘§)=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1š‘§1/šœ†1šœŽī‚µ1šœ†1ī‚¶.(2.21) The function š‘ž is convex and is the best dominant. The proof is complete.

Theorem 2.6. If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0, and 0ā‰¤š›¼<1, then one has š‘…šœ†š‘›,š‘š+11,šœ†2(š›¼)āŠ‚š¾šœ†š‘›,š‘š1,šœ†2(š›æ),(2.22) where š›æ=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1šœŽī‚µ1šœ†1ī‚¶,(2.23) and šœŽ is given by (2.15).

Proof. Let š‘“āˆˆš‘…šœ†š‘›,š‘š+11,šœ†2(š›¼), then from (2.9) we have ī‚€šœ‡Rešœ†š‘›,š‘š+11,šœ†2ī‚š‘“(š‘§)ī…ž>š›¼,(š‘§āˆˆš‘ˆ),(2.24) which is equivalent to ī‚€šœ‡šœ†š‘›,š‘š+11,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§.1+š‘§(2.25) Using Theorem 2.5, we have ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žā‰ŗš‘ž(š‘§)=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1š‘§1/šœ†1šœŽī‚µ1šœ†1ī‚¶.(2.26) Since š‘ž is convex and š‘ž(š‘ˆ) is symmetric with respect to the real axis, we deduce that ī‚ƒšœ‡Rešœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žī€·>Reš‘ž(1)=š›æ=š›æš›¼,šœ†1ī€ø=2š›¼āˆ’1+2(1āˆ’š›¼)šœ†1šœŽī‚µ1šœ†1ī‚¶,(2.27) from which we deduce š‘…šœ†š‘›,š‘š+11,šœ†2(š›¼)āŠ‚š¾šœ†š‘›,š‘š1,šœ†2(š›æ). This completes the proof of Theorem 2.6.

Remark 2.7. Special case of Theorem 2.6 with šœ†2=0 was given earlier in [11].

Theorem 2.8. Let š‘ž be a convex function in š‘ˆ, with š‘ž(0)=1, and let ā„Ž(š‘§)=š‘ž(š‘§)+šœ†1š‘§š‘žī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.28) If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0,and š‘“āˆˆš’œ and satisfies the differential subordination ī‚€šœ‡šœ†š‘›,š‘š+11,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(2.29) then ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ),(2.30) and this result is sharp.

Proof. Using (2.18) in (2.16), differential subordination (2.29) becomes š‘(š‘§)+šœ†1š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§)=š‘ž(š‘§)+šœ†1š‘§š‘žī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.31) Using Lemma 1.2, we obtain š‘(š‘§)ā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ).(2.32) Hence ī‚ƒšœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…žā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ).(2.33) And the result is sharp. This completes the proof of the theorem.

We give a simple application for Theorem 2.8.

Example 2.9. For š‘›=0,š‘š=1,šœ†2ā‰„šœ†1ā‰„0,š‘ž(š‘§)=(1+š‘§)/(1āˆ’š‘§),š‘“āˆˆš’œ, and š‘§āˆˆš‘ˆ and applying Theorem 2.8, we have ā„Ž(š‘§)=1+š‘§1āˆ’š‘§+šœ†1š‘§ī‚€1+š‘§ī‚1āˆ’š‘§ī…ž=1+2šœ†1š‘§āˆ’š‘§2(1āˆ’š‘§)2.(2.34) By using (1.8) we find that šœ‡šœ†0,11,šœ†2ī€·š‘“(š‘§)=1āˆ’šœ†1ī€øš‘“(š‘§)+šœ†1š‘§š‘“ī…ž(š‘§).(2.35) Now, šœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī€·(š‘§)=1āˆ’šœ†1ī€øīƒ¬š‘§+āˆžī“š‘˜=2š‘Žš‘˜š‘§š‘˜1+šœ†2īƒ­(š‘˜āˆ’1)+šœ†1īƒ¬š‘§+āˆžī“š‘˜=2š‘Žš‘˜š‘˜š‘§š‘˜1+šœ†2īƒ­.(š‘˜āˆ’1)(2.36) A straightforward calculation gives the following: ī‚ƒšœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž=1+āˆžī“š‘˜=2ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘˜š‘Žš‘˜1+šœ†2š‘§(š‘˜āˆ’1)š‘˜āˆ’1=āˆ‘š‘§+āˆžš‘˜=2ī€·š‘˜š‘Žš‘˜ī€·1+šœ†1ī€ø/ī€·(š‘˜āˆ’1)1+šœ†2š‘§(š‘˜āˆ’1)ī€øī€øš‘˜š‘§=ī€ŗš‘“(š‘§)āˆ—šœ™šœ†2ī€»āˆ—ī€ŗāˆ‘(š‘§)š‘§+āˆžš‘˜=2š‘˜ī€·1+šœ†1ī€øš‘§(š‘˜āˆ’1)š‘˜ī€»š‘§.(2.37) Similarly, using (1.8), we find that šœ‡šœ†0,21,šœ†2ī€·š‘“(š‘§)=1āˆ’šœ†1ī€øī‚ƒšœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)+šœ†1š‘§ī‚ƒšœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚„(š‘§)ī…ž,(2.38) then ī‚€šœ‡šœ†0,21,šœ†2ī‚š‘“(š‘§)ī…ž=ī‚€šœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚(š‘§)ī…ž+šœ†1š‘§ī‚€šœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2ī‚(š‘§)ī…žī…ž.(2.39) By using (2.37) we obtain ī‚€šœ‡šœ†0,11,šœ†2š‘“(š‘§)āˆ—šœ™šœ†2(ī‚š‘§)ī…žī…ž=āˆžī“š‘˜=2š‘˜(š‘˜āˆ’1)š‘Žš‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)1+šœ†2š‘§(š‘˜āˆ’1)š‘˜āˆ’2.(2.40) We get ī‚€šœ‡šœ†0,21,šœ†2ī‚š‘“(š‘§)ī…ž=1+āˆžī“š‘˜=2š‘˜š‘Žš‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)1+šœ†2š‘§(š‘˜āˆ’1)š‘˜āˆ’1+šœ†1āˆžī“š‘˜=2š‘˜(š‘˜āˆ’1)š‘Žš‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)1+šœ†2š‘§(š‘˜āˆ’1)š‘˜āˆ’1=1+āˆžī“š‘˜=2š‘˜š‘Žš‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)21+šœ†2š‘§(š‘˜āˆ’1)š‘˜āˆ’1=ī€ŗš‘“(š‘§)āˆ—šœ™šœ†2ī€»āˆ—ī‚ƒāˆ‘(š‘§)š‘§+āˆžš‘˜=2š‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)2š‘§š‘˜ī‚„š‘§.(2.41) From Theorem 2.8 we deduce that ī€ŗš‘“(š‘§)āˆ—šœ™šœ†2ī€»āˆ—ī‚ƒāˆ‘(š‘§)š‘§+āˆžš‘˜=2š‘˜ī€·1+šœ†1ī€ø(š‘˜āˆ’1)2š‘§š‘˜ī‚„š‘§ā‰ŗ1+2šœ†1š‘§āˆ’š‘§2(1āˆ’š‘§)2,(2.42) implies that ī€ŗš‘“(š‘§)āˆ—šœ™šœ†2ī€»āˆ—ī€ŗāˆ‘(š‘§)š‘§+āˆžš‘˜=2š‘˜ī€·1+šœ†1ī€øš‘§(š‘˜āˆ’1)š‘˜ī€»š‘§ā‰ŗ1+š‘§,1āˆ’š‘§(š‘§āˆˆš‘ˆ).(2.43)

Theorem 2.10. Let š‘ž be a convex function in š‘ˆ, with š‘ž(0)=1 and let ā„Ž(š‘§)=š‘ž(š‘§)+š‘§š‘žī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.44) If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0, and š‘“āˆˆš’œ and satisfies the differential subordination ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§),(2.45) then šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§ā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ).(2.46) And the result is sharp.

Proof. Let šœ‡š‘(š‘§)=šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§=āˆ‘š‘§+āˆžš‘˜=2ī‚€ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘š/ī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šāˆ’1ī‚š‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜š‘§=1+š‘1š‘§+š‘2š‘§2[]+ā‹Æ,(š‘āˆˆā„‹1,1,š‘§āˆˆš‘ˆ).(2.47) Differentiating (2.47), with respect to š‘§, we obtain ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…ž=š‘(š‘§)+š‘§š‘ī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.48) Using (2.48), (2.45) becomes š‘(š‘§)+š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§)=š‘ž(š‘§)+š‘§š‘žī…ž(š‘§).(2.49) Using Lemma 1.2, we deduce that š‘(š‘§)ā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ),(2.50) and using (2.47), we have šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§ā‰ŗš‘ž(š‘§),(š‘§āˆˆš‘ˆ).(2.51) This proves Theorem 2.10.

We give a simple application for Theorem 2.10.

Example 2.11. For š‘›=0,š‘š=1,šœ†2ā‰„šœ†1ā‰„0,š‘ž(š‘§)=1/(1āˆ’š‘§),š‘“āˆˆš’œ, and š‘§āˆˆš‘ˆ and applying Theorem 2.10, we have 1ā„Ž(š‘§)=ī‚€11āˆ’š‘§+š‘§ī‚1āˆ’š‘§ī…ž=1(1āˆ’š‘§)2.(2.52) From Example 2.9, we have šœ‡šœ†0,11,šœ†2ī€·š‘“(š‘§)=1āˆ’šœ†1ī€øš‘“(š‘§)+šœ†1š‘§š‘“ā€²(š‘§),(2.53) so ī‚€šœ‡šœ†0,11,šœ†2ī‚š‘“(š‘§)ī…ž=š‘“ā€²(š‘§)+šœ†1š‘§š‘“ī…žī…ž(š‘§).(2.54) Now, from Theorem 2.10 we deduce that š‘“ā€²(š‘§)+šœ†1š‘§š‘“ī…žī…ž1(š‘§)ā‰ŗ(1āˆ’š‘§)2(2.55) implies that ī€·1āˆ’šœ†1ī€øš‘“(š‘§)+šœ†1š‘§š‘“ā€²(š‘§)š‘§ā‰ŗ1.1āˆ’š‘§(2.56)

Theorem 2.12. Let ā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§1+š‘§,(š‘§āˆˆš‘ˆ),(2.57) be convex in š‘ˆ, with ā„Ž(0)=1 and 0ā‰¤š›¼<1. If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0,š‘“āˆˆš’œ, and the differential subordination holds as ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§),(2.58) then šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§ā‰ŗš‘ž(š‘§)=2š›¼āˆ’1+2(1āˆ’š›¼)ln(1+š‘§)š‘§.(2.59) The function š‘ž is convex and is the best dominant.

Proof. Let šœ‡š‘(š‘§)=šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§=āˆ‘š‘§+āˆžš‘˜=2ī‚€ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘š/ī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šāˆ’1ī‚š‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜š‘§=1+š‘1š‘§+š‘2š‘§2[]+ā‹Æ,(š‘āˆˆā„‹1,1,š‘§āˆˆš‘ˆ).(2.60) Differentiating (2.60), with respect to š‘§, we obtain ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…ž=š‘(š‘§)+š‘§š‘ī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.61) Using (2.61), the differential subordination (2.58) becomes š‘(š‘§)+š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§1+š‘§,(š‘§āˆˆš‘ˆ).(2.62) From Lemma 1.1, we deduce that 1š‘(š‘§)ā‰ŗš‘ž(š‘§)=š‘§ī€œš‘§0=1ā„Ž(š‘”)š‘‘š‘”š‘§ī€œš‘§0ī‚µ1+(2š›¼āˆ’1)š‘”ī‚¶=11+š‘”š‘‘š‘”š‘§ī‚øī€œš‘§01ī€œ1+š‘”š‘‘š‘”+(2š›¼āˆ’1)š‘§0š‘”ī‚¹1+š‘”š‘‘š‘”=2š›¼āˆ’1+2(1āˆ’š›¼)ln(1+š‘§)š‘§.(2.63) Using (2.60), we have šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§ā‰ŗš‘ž(š‘§)=2š›¼āˆ’1+2(1āˆ’š›¼)ln(1+š‘§)š‘§.(2.64) The proof is complete.

From Theorem 2.12, we deduce the following corollary.

Corollary 2.13. If š‘“āˆˆš‘…šœ†š‘›,š‘š1,šœ†2(š›¼), then īƒ©šœ‡Rešœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§īƒŖ>(2š›¼āˆ’1)+2(1āˆ’š›¼)ln2,(š‘§āˆˆš‘ˆ).(2.65)

Proof. Since š‘“āˆˆš‘…šœ†š‘›,š‘š1,šœ†2(š›¼), from Definition 2.3 we have ī‚€šœ‡Rešœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…ž>š›¼,(š‘§āˆˆš‘ˆ),(2.66) which is equivalent to ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§)=1+(2š›¼āˆ’1)š‘§.1+š‘§(2.67) Using Theorem 2.12, we have šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§ā‰ŗš‘ž(š‘§)=(2š›¼āˆ’1)+2(1āˆ’š›¼)ln(1+š‘§)š‘§.(2.68) Since š‘ž is convex and š‘ž(š‘ˆ) is symmetric with respect to the real axis, we deduce that īƒ©šœ‡Rešœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§īƒŖ>Reš‘ž(1)=(2š›¼āˆ’1)+2(1āˆ’š›¼)ln2,(š‘§āˆˆš‘ˆ).(2.69)

Theorem 2.14. Let ā„Žāˆˆā„‹(š‘ˆ), with ā„Ž(0)=1, ā„Žā€²(0)ā‰ 0 which satisfy the inequality ī‚µRe1+š‘§ā„Žī…žī…žī…ž(š‘§)ā„Žī…ž(ī‚¶1š‘§)>āˆ’2,(š‘§āˆˆš‘ˆ).(2.70) If š‘›,š‘šāˆˆā„•0,šœ†2ā‰„šœ†1ā‰„0,and š‘“āˆˆš’œ and satisfies the differential subordination ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…žā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ),(2.71) then šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§1ā‰ŗš‘ž(š‘§)=š‘§ī€œš‘§0ā„Ž(š‘”)š‘‘š‘”.(2.72)

Proof. Let šœ‡š‘(š‘§)=šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§=āˆ‘š‘§+āˆžš‘˜=2ī‚€ī€·1+šœ†1ī€ø(š‘˜āˆ’1)š‘š/ī€·1+šœ†2ī€ø(š‘˜āˆ’1)š‘šāˆ’1ī‚š‘(š‘›,š‘˜)š‘Žš‘˜š‘§š‘˜š‘§=1+š‘1š‘§+š‘2š‘§2[]+ā‹Æ,(š‘āˆˆā„‹1,1,š‘§āˆˆš‘ˆ).(2.73) Differentiating (2.73), with respect to š‘§, we have ī‚€šœ‡šœ†š‘›,š‘š1,šœ†2ī‚š‘“(š‘§)ī…ž=š‘(š‘§)+š‘§š‘ī…ž(š‘§),(š‘§āˆˆš‘ˆ).(2.74) Using (2.74), the differential subordination (2.71) becomes š‘(š‘§)+š‘§š‘ī…ž(š‘§)ā‰ŗā„Ž(š‘§),(š‘§āˆˆš‘ˆ).(2.75) From Lemma 1.1, we deduce that 1š‘(š‘§)ā‰ŗš‘ž(š‘§)=š‘§ī€œš‘§0ā„Ž(š‘”)š‘‘š‘”,(2.76) and using (2.73), we obtain šœ‡šœ†š‘›,š‘š1,šœ†2š‘“(š‘§)š‘§1ā‰ŗš‘ž(š‘§)=š‘§ī€œš‘§0ā„Ž(š‘”)š‘‘š‘”.(2.77) From Lemma 1.3, we have that the function š‘ž is convex, and from Lemma 1.1, š‘ž is the best dominant for subordination (2.71). This completes the proof of Theorem 2.14.

3. Conclusion

We remark that several subclasses of analytic univalent functions can be derived and studied using the operator šœ‡šœ†š‘›,š‘š1,šœ†2.

Acknowledgment

This work is fully supported by UKM-ST-06-FRGS0107-2009, MOHE, Malaysia.

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Copyright © 2010 Ma'moun Harayzeh Al-Abbadi and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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