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Advances in Decision Sciences
VolumeΒ 2011, Article IDΒ 167672, 9 pages
http://dx.doi.org/10.1155/2011/167672
Research Article

Majorization for A Subclass of 𝛽-Spiral Functions of Order 𝛼 Involving a Generalized Linear Operator

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia
2Department of Mathematics-Informatics, Faculty of Sciences, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania

Received 22 June 2011; Accepted 18 August 2011

Academic Editor: SheltonΒ Peiris

Copyright Β© 2011 Afaf A. Ali Abubaker et al. 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.

Abstract

Motivated by Carlson-Shaffer linear operator, we define here a new generalized linear operator. Using this operator, we define a class of analytic functions in the unit disk π‘ˆ. For this class, a majorization problem of analytic functions is discussed.

1. Introduction

Let 𝐴 denote the class of functions 𝑓(𝑧) of the form𝑓(𝑧)=𝑧+βˆžξ“π‘›=1π‘Žπ‘›+1𝑧𝑛+1(1.1) which are analytic in the unit disk π‘ˆ={π‘§βˆˆβ„‚βˆΆ|𝑧|<1}.

Let 𝑓 and 𝑔 be analytic in π‘ˆ. Then, we say that function 𝑓 is subordinate to 𝑔 if there exists a Schwarz function πœ”(𝑧), analytic in π‘ˆ with πœ”(0)=0 and |πœ”(𝑧)|<1, such that 𝑓(𝑧)=𝑔(πœ”(𝑧)), π‘§βˆˆπ‘ˆ (see [1]). We denote this subordination by𝑓≺𝑔(π‘§βˆˆπ‘ˆ).(1.2)

Further, 𝑓 is said to be quasi subordinate to 𝑔 if there exists an analytic function πœ‘(𝑧) such that 𝑓(𝑧)/πœ‘(𝑧) is analytic in π‘ˆ,𝑓(𝑧)πœ‘(𝑧)≺𝑔(𝑧),(π‘§βˆˆπ‘ˆ)(1.3) and |πœ‘(𝑧)|≀1. Note that the quasi subordination (1.3) is equivalent to𝑓(𝑧)=πœ‘(𝑧)𝑔(πœ”(𝑧)),(1.4) where |πœ‘(𝑧)|≀1 and |πœ”(𝑧)|≀|𝑧|<1 (see [2]). If πœ‘(𝑧)=1, then (1.3) becomes (1.2).

Let functions 𝑓 and 𝑔 be analytic functions in π‘ˆ. If |𝑓(𝑧)|≀|𝑔(𝑧)|, then there exists a function πœ‘ analytic in π‘ˆ such that |πœ‘(𝑧)|≀1 in π‘ˆ, for which𝑓(𝑧)=πœ‘(𝑧)𝑔(𝑧)(π‘§βˆˆπ‘ˆ).(1.5) In this case, we say that 𝑓 is majorized by 𝑔 in π‘ˆ (see [3]), and we write𝑓(𝑧)β‰ͺ𝑔(𝑧)(π‘§βˆˆπ‘ˆ).(1.6) If we take πœ”(𝑧)=𝑧 in (1.4), then the quasi subordination (1.3) becomes the majorization (1.6).

Also, let 𝑆 denote the subclass of 𝐴 consisting of all functions which are univalent in π‘ˆ.

In [4], Robertson introduced star-like functions of order 𝛼 on π‘ˆ.

Definition 1.1. Let 0≀𝛼<1 and π‘“βˆˆπ΄; then, 𝑓 is a star-like function of order 𝛼 on π‘ˆ if and only if β„œξ‚»π‘§π‘“β€²(𝑧)𝑓(𝑧)>𝛼(π‘§βˆˆπ‘ˆ).(1.7) Let π‘†βˆ—(𝛼) denote the whole star-like functions of order 𝛼 in π‘ˆ.

Spaček [5] extended the class of π‘†βˆ— and obtained the class of 𝛽-spiral-like functions. In the same article, the author gave an analytical characterization of spirallikeness of type 𝛽 on π‘ˆ.

Definition 1.2. Let βˆ’πœ‹/2<𝛽<πœ‹/2 and π‘“βˆˆπ΄; then, 𝑓 is 𝛽-spiral-like function on π‘ˆ if and only if β„œξ‚»π‘’π‘–π›½π‘“β€²(𝑧)𝑓(𝑧)>0(π‘§βˆˆπ‘ˆ).(1.8) We denote the whole 𝛽-spiral-like functions in π‘ˆ by π‘†βˆ—π›½.

Finally, Libera [6] introduced and studied the class of 𝛽-spiral-like functions of order 𝛼.

Definition 1.3. Let 0≀𝛼<1, βˆ’πœ‹/2<𝛽<πœ‹/2 and π‘“βˆˆπ΄; then, 𝑓 is 𝛽-spiral function of order 𝛼 if and only if β„œξ‚»π‘’π‘–π›½π‘§π‘“β€²(𝑧)𝑓(𝑧)>𝛼cos𝛽(π‘§βˆˆπ‘ˆ).(1.9) We denote the whole 𝛽-spiral-like functions of order 𝛼 in π‘ˆ by π‘†βˆ—π›½(𝛼).

In particular, we consider the convolution with function πœ™(π‘Ž,𝑐) defined by 𝐿(π‘Ž,𝑏)𝑓(𝑧)=𝑧+βˆžξ“π‘›=1(π‘Ž)𝑛(𝑐)𝑛𝑧𝑛+1,(1.10) where π‘Žβˆˆβ„‚, 𝑏≠0,βˆ’1,βˆ’2,…, and (π‘Ž)𝑛 is the Pochhammer symbol defined by (π‘Ž)𝑛=Ξ“(π‘Ž+𝑛)=ξ‚»Ξ“(π‘Ž)1,𝑛=0,π‘Ž(π‘Ž+1)β‹―(π‘Ž+π‘›βˆ’1),𝑛={1,2,3,…}.(1.11) Function πœ™(π‘Ž,𝑐) is an incomplete beta-function related to the Gauss hypergeometric function by πœ™(π‘Ž,𝑐;𝑧)=𝑧2𝐹1(1,π‘Ž;𝑐;𝑧).(1.12) It has an analytic continuation to the 𝑧-plane cut along the positive real line from 1 to ∞. We note that πœ™(π‘Ž.1;𝑧)=𝑧/(1βˆ’π‘§)π‘Ž and πœ™(2,1;𝑧) are the Koebe functions.

Carlson and Shaffer [7] defined a convolution operator on 𝐴 involving an incomplete beta-function as 𝐿(π‘Ž,𝑏)𝑓(𝑧)=πœ™(π‘Ž,𝑐;𝑧)βˆ—π‘“(𝑧)=𝑧+βˆžξ“π‘›=1(π‘Ž)𝑛(𝑐)π‘›π‘Žπ‘›+1𝑧𝑛+1.(1.13)

Definition 1.4. Let function 𝐹 be given by 𝐹(π‘š,β„“,πœ†)=βˆžξ“π‘›=0ξ‚€1+β„“+πœ†π‘›ξ‚1+β„“π‘šπ‘§π‘›+1,(1.14) where β„“,πœ†β‰₯0 and π‘šβˆˆβ„€. The generalized linear operator 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)βˆΆπ΄β†’π΄ is given as 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑏)𝑓(𝑧)=𝑧+βˆžξ“π‘›=1ξ‚€1+β„“+πœ†π‘›ξ‚1+β„“π‘š(π‘Ž)𝑛(𝑐)π‘›π‘Žπ‘›+1𝑧𝑛+1.(1.15)

We note here some special cases.(1)𝐿(0,β„“,πœ†,π‘Ž,𝑏)𝑓(𝑧)=𝐿(π‘Ž,𝑏)𝑓(𝑧) is the Carlson and Shaffer operator [7]. (2)𝐿(0,β„“,πœ†,𝛿+1,1)𝑓(𝑧),π›Ώβˆˆβ„•0, is the Ruscheweyh derivative [8]. (3)𝐿(π‘š,0,πœ†,1,1)𝑓(𝑧),π‘šβˆˆβ„•0, is the Al-Oboudi operator [9]. (4)𝐿(π‘š,0,πœ†,π‘Ž,𝑏)𝑓(𝑧) is the linear operator introduced by Al-Refai and Darus [10]. (5)𝐹(π‘š,β„“,πœ†),π‘šβˆˆβ„•0, is the generalized multiplier transformation which was introduced and studied by CΓ‘tÑş [11].(6)𝐹(π‘š,β„“,1),π‘šβˆˆβ„•0, is the multiplier transformation which was introduced and studied by Cho and Srivastava [12] and Cho and Kim [13].

Remark 1.5. It follows from the above definition that 𝑧(𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓(𝑧))β€²=π‘ŽπΏ(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)βˆ’(π‘Žβˆ’1)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓(𝑧)(π‘§βˆˆπ‘ˆ).(1.16)

We introduce the class π‘†βˆ—π›½(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼) as follows.

Definition 1.6. Let π‘Žβˆˆβ„‚, 𝑐≠0,βˆ’1,βˆ’2,…,β„“, πœ†β‰₯0, π‘šβˆˆβ„€, 0≀𝛼<1, βˆ’πœ‹/2<𝛽<πœ‹/2, and π‘“βˆˆπ΄; then, one hasπ‘†βˆ—π›½(π‘š,β„“,πœ†,π‘Ž,𝑐,β„“,πœ†,𝛼) if and only if β„œξ‚»π‘’π‘–π›½π‘§(𝐿(π‘Ž,𝑐)𝑓(𝑧))′𝐿(π‘Ž,𝑐)𝑓(𝑧)>𝛼cos𝛽.(1.17) Obviously, when π‘Ž=𝑐=1 and π‘š=0 we obtain π‘“βˆˆπ‘†βˆ—π›½(𝛼), when π‘Ž=𝑐=1 and π‘š=𝛽=0, we obtain that 𝑓(𝑧) is a starl-like function of order 𝛼 on π‘ˆ, and also when π‘Ž=𝑐=1 and π‘š=𝛼=0, we obtain that 𝑓(𝑧) is spiral-like function of type 𝛽 on π‘ˆ.

Biernacki [14] in 1936 obtained the first results of majorization-subordination theory. He showed that, if 𝑔(𝑧)βˆˆπ‘† and 𝑓(𝑧)≺𝑔(𝑧) in π‘ˆ, then 𝑓(𝑧)β‰ͺ𝑔(𝑧) in |𝑧|≀(1/4). Goluzin [15] improved the result and Shah [16] obtained the complete solution for 𝑆 by showing that 𝑓(𝑧)β‰ͺ𝑔(𝑧) in √|𝑧|≀(3βˆ’5)/2 and that the result is the best possible. A majorization problem for star-like functions has been given by MacGregor [3]. Also, majorization problem for star-like functions of complex order has recently been investigated by Altintaş et al. [17].

The main object of this paper is to investigate the problem of majorization of the class π‘†βˆ—π›½(β„“,πœ†,π‘Ž,𝑐,𝛼) defined by a generalized linear operator.

In order to prove our main theorem we need the following lemma.

Lemma 1.7 (see [18]). Let πœ‘(𝑧) be analytic in π‘ˆ satisfying |πœ‘(𝑧)|≀1 for π‘§βˆˆπ‘ˆ. Then, ||||≀||||πœ‘β€²(𝑧)1βˆ’πœ‘(𝑧)21βˆ’|𝑧|2.(1.18)

2. Main Results

Theorem 2.1. Let function π‘“βˆˆπ΄ and suppose that π‘”βˆˆπ‘†βˆ—π›½(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼). If 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓 is majorized by 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔 in π‘ˆ, then ||𝐿||≀||𝐿||ξ€·(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)|𝑧|β‰€π‘Ÿ1ξ€Έ,(2.1) where π‘Ÿ1||=π‘Ÿ(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼,𝛽)=2+|π‘Ž|+2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||2||2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||βˆ’βˆšΞ˜(π‘Ž,𝛼,𝛽)2||2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||,(2.2)Θ(π‘Ž,𝛼,𝛽)=4+|π‘Ž|2+||2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||2||+4|π‘Ž|+42(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||||βˆ’2|π‘Ž|2(1βˆ’π›Όcos𝛽)βˆ’π‘Žπ‘’π‘–π›½||,(2.3) for π‘Žβˆˆβ„‚,𝑐≠0,βˆ’1,βˆ’2,…,,β„“,πœ†β‰₯0,π‘šβˆˆβ„€,0≀𝛼<1,βˆ’πœ‹/2<𝛽<πœ‹/2, and |π‘Ž|β‰₯|2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½|.

Proof. Since π‘”βˆˆπ‘†βˆ—π›½(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼), we have 𝑒𝑖𝛽𝑧(𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧))β€²=𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)1+(1βˆ’2𝛼)πœ”1βˆ’πœ”cos𝛽+𝑖sin𝛽,(2.4) where πœ” is analytic in π‘ˆ, with πœ”(0)=0 and |πœ”|≀|𝑧|<1(π‘§βˆˆπ‘ˆ).(2.5) By using (1.16) in (2.4), we get 𝑒𝑖𝛽[]π‘ŽπΏ(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)βˆ’(π‘Žβˆ’1)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)=𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)1+(1βˆ’2𝛼)πœ”1βˆ’πœ”cos𝛽+𝑖sin𝛽.(2.6) Hence, 𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)=𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)π‘Žπ‘’π‘–π›½+ξ€·2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½ξ€Έπœ”π‘Žπ‘’π‘–π›½,(1βˆ’πœ”)(2.7) which, in view of (2.5), immediately yields the inequality ||||≀||𝑒𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)𝑖𝛽|||π‘Ž|(1+|𝑧|)|||π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||||||.|𝑧|𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)(2.8) Next, since 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓 is majorized by 𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔 in π‘ˆ, from (1.5) we have 𝑧(𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓(𝑧))β€²=π‘§πœ‘ξ…ž(𝑧)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)+π‘§πœ‘(𝑧)(𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧))β€².(2.9) Also, by using (1.16) in (2.11), we get π‘ŽπΏ(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)βˆ’(π‘Žβˆ’1)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑓(𝑧)=π‘§πœ‘ξ…ž[];(𝑧)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)+πœ‘(𝑧)π‘ŽπΏ(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)βˆ’(π‘Žβˆ’1)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)(2.10) then, we have 1𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)=π‘Žπ‘§πœ‘ξ…ž(𝑧)𝐿(π‘š,β„“,πœ†,π‘Ž,𝑐)𝑔(𝑧)+πœ‘(𝑧)𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧).(2.11) Thus, by Lemma 1.7, since the Schwarz function πœ™ satisfies the inequality in (1.18) and using (2.8) in (2.11), we get ||𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(z)||≀||||1βˆ’πœ‘(𝑧)2|𝑧|ξ€·||(1βˆ’|𝑧|)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||ξ€ΈΓ—||||+||||||||.|𝑧|𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)πœ‘(𝑧)𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)(2.12) Hence, ||||≀||||𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)1βˆ’πœ‘(𝑧)2|||𝑧|+(1βˆ’|𝑧|)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||ξ€Έ|||||𝑧|πœ‘(𝑧)ξ€·||(1βˆ’|𝑧|)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||ξ€ΈΓ—||||,|𝑧|𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)(2.13) which, upon setting |||||𝑧|=π‘Ÿ,πœ‘(𝑧)=𝜌(0β‰€πœŒβ‰€1)(2.14) yields ||||≀𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑓(𝑧)πœƒ(𝜌)ξ€·||(1βˆ’π‘Ÿ)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿξ€Έ||||,𝐿(π‘š,β„“,πœ†,π‘Ž+1,𝑐)𝑔(𝑧)(2.15) where function πœƒ(𝜌) defined by ξ€·πœƒ(𝜌)=1βˆ’πœŒ2ξ€Έξ€·||π‘Ÿ+(1βˆ’π‘Ÿ)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿξ€ΈπœŒ(2.16) takes on its maximum value at 𝜌=1 with π‘Ÿ1[][]=π‘Ÿ(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼,𝛽)=max{π‘Ÿβˆˆ0,1βˆΆπœ“(π‘Ÿ,𝜌)≀1,βˆ€πœŒβˆˆ0,1},(2.17) where πœ“(π‘Ÿ,𝜌)=πœƒ(𝜌)ξ€·||(1βˆ’π‘Ÿ)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿξ€Έ;(2.18) then, we have πœƒ(𝜌)ξ€·||(1βˆ’π‘Ÿ)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿξ€Έβ‰€1.(2.19) A simple calculus in (2.19) is equivalent to ξ€·||βˆ’(1+𝜌)π‘Ÿ+(1βˆ’π‘Ÿ)|π‘Ž|βˆ’2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿξ€Έβ‰₯0,(2.20) while the inequality in (2.19) takes its minimum value at 𝜌=1, that is, ||2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||π‘Ÿ2βˆ’ξ€·||2|π‘Ž|+2(1βˆ’π›Ό)cosπ›½βˆ’π‘Žπ‘’π‘–π›½||ξ€Έπ‘Ÿ+|π‘Ž|β‰₯0,(2.21) for all π‘Ÿβˆˆ[0,π‘Ÿ1], where π‘Ÿ1=π‘Ÿ(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼,𝛽) given in (2.2) holds true for |𝑧|β‰€π‘Ÿ(π‘š,β„“,πœ†,π‘Ž,𝑐,𝛼,𝛽), which proves the conclusion (2.1).

Putting π‘š=𝛼=𝛽=0 in Theorem 2.1, we obtain the following result.

Corollary 2.2. Let function π‘“βˆˆπ΄ and suppose that π‘”βˆˆπ‘†βˆ—(π‘Ž,𝑐). If 𝐿(π‘Ž,𝑐)𝑓 is majorized by 𝐿(π‘Ž,𝑐)𝑔 in π‘ˆ, then ||𝐿||≀||𝐿||ξ€·(π‘Ž+1,𝑐)𝑓(𝑧)(π‘Ž+1,𝑐)𝑔(𝑧)|𝑧|β‰€π‘Ÿ2ξ€Έ,=π‘Ÿ(π‘Ž,𝑐)(2.22) where ||||π‘Ÿ(π‘Ž,𝑐)=3+|π‘Ž|+2βˆ’π‘Ž2||||βˆ’ξ”2βˆ’π‘Ž||||4+2βˆ’π‘Ž2||||βˆ’2|π‘Ž|2βˆ’π‘Ž+4|π‘Ž|+|π‘Ž|22||||.2βˆ’π‘Ž(2.23)

Further, putting π‘Ž=𝑐=1 and π‘š=0 in Theorem 2.1, we obtain the result of Altintaş et al. [17].

Corollary 2.3. Let function π‘“βˆˆπ΄ and suppose that π‘”βˆˆπ‘†βˆ—((π›Όβˆ’1)𝑒𝑖𝛽)=π‘†βˆ—π›½(𝛼), where 0≀𝛼<1 and βˆ’πœ‹/2<𝛽<πœ‹/2. If 𝑓 is majorized by 𝑔 in π‘ˆ, then ||π‘“ξ…ž||≀||𝑔(𝑧)ξ…ž||ξ€·(𝑧)|𝑧|β‰€π‘Ÿ3ξ€Έ,=π‘Ÿ(𝛼,𝛽)(2.24) where ||π‘Ÿ(𝛼,𝛽)=3+2(π›Όβˆ’1)𝑒𝑖𝛽||βˆ’12||2(π›Όβˆ’1)𝑒𝑖𝛽||βˆ’ξ”βˆ’1||9+2(π›Όβˆ’1)𝑒𝑖𝛽||βˆ’12||||+22(π›Όβˆ’1)βˆ’12||2(π›Όβˆ’1)𝑒𝑖𝛽||.βˆ’1(2.25)

Putting 𝛽=0 in Corollary 2.3, we obtain the result as follows.

Corollary 2.4. Let function π‘“βˆˆπ΄ and suppose that π‘”βˆˆπ‘†βˆ—(𝛼), where 0≀𝛼<1. If 𝑓 is majorized by 𝑔 in π‘ˆ, then ||||≀||||𝑓′(𝑧)𝑔′(𝑧)|𝑧|β‰€π‘Ÿ4ξ€Έ,=π‘Ÿ(𝛼)(2.26) where ||||π‘Ÿ(𝛼)=3+1βˆ’2𝛼2||||βˆ’ξ”1βˆ’2𝛼||||9+1βˆ’2𝛼2||||+22(π›Όβˆ’1)βˆ’12||||.1βˆ’2𝛼(2.27)

Also, putting 𝛼=𝛽=0 in Corollary 2.3, we obtain the result of MacGregor [3].

Corollary 2.5. Let function π‘“βˆˆπ΄ and suppose that π‘”βˆˆπ‘†βˆ—(0). If 𝑓 is majorized by 𝑔 in π‘ˆ, then ||π‘“ξ…ž||≀||𝑔(𝑧)ξ…ž||ξ‚€βˆš(𝑧)|𝑧|≀2βˆ’3.(2.28)

Acknowledgment

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.

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