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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 837913, 10 pages
Research Article

On Certain Sufficiency Criteria for 𝑝-Valent Meromorphic Spiralike Functions

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan

Received 10 June 2012; Accepted 6 August 2012

Academic Editor: AllanΒ Peterson

Copyright Β© 2012 Muhammad Arif. 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.


We consider some subclasses of meromorphic multivalent functions and obtain certain simple sufficiency criteria for the functions belonging to these classes. We also study the mapping properties of these classes under an integral operator.

1. Introduction

Let βˆ‘(𝑝,𝑛) denote the class of functions 𝑓(𝑧) of the form 𝑓(𝑧)=π‘§βˆ’π‘+βˆžξ“π‘˜=π‘›π‘Žπ‘˜π‘§π‘˜βˆ’π‘+1(π‘βˆˆβ„•),(1.1) which are analytic and 𝑝-valent in the punctured unit disk π•Œ={π‘§βˆΆ0<|𝑧|<1}. Also let βˆ‘βˆ—πœ†(𝑝,𝑛,𝛼) and βˆ‘πœ†π‘(𝑝,𝑛,𝛼) denote the subclasses of βˆ‘(𝑝,𝑛) consisting of all functions 𝑓(𝑧) which are defined, respectively, by ξ‚΅Reβˆ’π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)𝑓(𝑧)>𝛼cosπœ†(π‘§βˆˆπ•Œ),Reβˆ’π‘’π‘–πœ†ξ€·π‘§π‘“ξ…žξ€Έ(𝑧)ξ…žπ‘“ξ…žξƒͺ(𝑧)>𝛼cosπœ†(π‘§βˆˆπ•Œ).(1.2)

We note that for πœ†=0 and 𝑛=1, the above classes reduce to the well-known subclasses of βˆ‘(𝑝) consisting of meromorphic multivalent functions which are, respectively, starlike and convex of order 𝛼 (0≀𝛼<𝑝). For the detail on the subject of meromorphic spiral-like functions and related topics, we refer the work of Liu and Srivastava [1], Goyal and Prajapat [2], Raina and Srivastava [3], Xu and Yang [4], and Spacek [5] and Robertson [6].

Analogous to the subclass of βˆ‘(1,1) for meromorphic univalent functions studied by Wang et al. [7] and Nehari and Netanyahu [8], we define a subclass βˆ‘πœ†π‘(𝑝,𝑛,𝛼) of βˆ‘(𝑝,𝑛) consisting of functions 𝑓(𝑧) satisfying βˆ’Reπ‘’π‘–πœ†ξƒ©ξ€·π‘§π‘“ξ…žξ€Έ(𝑧)ξ…žπ‘“ξ…žξƒͺ(𝑧)<𝛼cosπœ†(𝛼>𝑝,π‘§βˆˆπ•Œ).(1.3)

For more details of the above classes see also [9, 10].

Motivated from the work of Frasin [11], we introduce the following integral operator of multivalent meromorphic functions βˆ‘(𝑝)π»π‘š,𝑝1(𝑧)=𝑧𝑝+1ξ€œπ‘§0π‘šξ‘π‘—=1𝑒𝑝𝑓𝑗(𝑒)𝛼𝑗𝑑𝑒.(1.4)

For 𝑝=1, (1.4) reduces to the integral operator introduced and studied by Mohammed and Darus [12, 13]. Similar integral operators for different classes of analytic, univalent, and multivalent functons in the open unit disk are studied by various authors, see [14–19].

In this paper, first, we find sufficient conditions for the classes βˆ‘βˆ—πœ†(𝑝,𝑛,𝛼) and βˆ‘πœ†π‘(𝑝,𝑛,𝛼) and then study some mapping properties of the integral operator given by (1.4).

We will assume throughout our discussion, unless otherwise stated, that πœ† is real with |πœ†|<πœ‹/2, 0≀𝛼<𝑝, 𝑝,π‘›βˆˆβ„•, 𝛼𝑗>0 for π‘—βˆˆ{1,β€¦π‘š}.

To obtain our main results, we need the following Lemmas.

Lemma 1.1 (see [20]). If βˆ‘π‘ž(𝑧)∈(1,𝑛) with 𝑛β‰₯1 and satisfies the condition ||𝑧2π‘žξ…ž||<𝑛(𝑧)+1βˆšπ‘›2+1(π‘§βˆˆπ•Œ),(1.5) then π‘ž(𝑧)βˆˆβˆ—ξ“0(1,𝑛,0).(1.6)

Lemma 1.2 (see [21]). Let Ξ© be a set in the complex plane β„‚ and suppose that Ξ¨ is a mapping from β„‚2Γ—π•Œ to β„‚ which satisfies Ξ¨(𝑖π‘₯,𝑦,𝑧)βˆ‰Ξ© for π‘§βˆˆπ•Œ, and for all real π‘₯,𝑦 such that 𝑦≀(βˆ’π‘›/2)(1+π‘₯2). If β„Ž(𝑧)=1+𝑐𝑛𝑧𝑛+β‹― is analytic in π•Œ and Ξ¨(β„Ž(𝑧),π‘§β„Žξ…ž(𝑧),𝑧)∈Ω for all π‘§βˆˆπ•Œ, then Reβ„Ž(𝑧)>0.

2. Some Properties of the Classes βˆ‘βˆ—πœ†(𝑝,𝑛,𝛼) and βˆ‘πœ†π‘(𝑝,𝑛,𝛼)

Theorem 2.1. If βˆ‘π‘“(𝑧)∈(𝑝,𝑛) satisfies ||||(𝑧𝑝𝑓(𝑧))π‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†ξ‚»π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)ξ‚Ό+||||<𝑛𝑓(𝑧)+𝛼cosπœ†+𝑖𝑝sinπœ†(π‘βˆ’π›Ό)cosπœ†βˆšπ‘›2(+1π‘βˆ’π›Ό)cosπœ†(π‘§βˆˆπ•Œ),(2.1) then βˆ‘π‘“(𝑧)βˆˆβˆ—πœ†(𝑝,𝑛,𝛼).

Proof. Let us set a function β„Ž(𝑧) by 1β„Ž(𝑧)=𝑧(𝑧𝑝𝑓(𝑧))π‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†=1𝑧+π‘’π‘–πœ†π‘Žπ‘›π‘§(π‘βˆ’π›Ό)cosπœ†π‘›+β‹―(2.2) for βˆ‘π‘“(𝑧)∈(𝑝,𝑛). Then clearly (2.2) shows that βˆ‘β„Ž(𝑧)∈(1,𝑛).
Differentiating (2.2) logarithmically, we have β„Žξ…ž(𝑧)=π‘’β„Ž(𝑧)π‘–πœ†ξ‚Έπ‘“(π‘βˆ’π›Ό)cosπœ†ξ…ž(𝑧)+𝑝𝑓(𝑧)π‘§ξ‚Ήβˆ’1𝑧(2.3) which gives ||𝑧2β„Žξ…ž||=||||(𝑧)+1(𝑧𝑝𝑓(𝑧))π‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†1𝑒(π‘βˆ’π›Ό)cosπœ†π‘–πœ†π‘§π‘“ξ…ž(𝑧)𝑓||||.(𝑧)+𝛼cosπœ†+𝑖𝑝sinπœ†+1(2.4)
Thus using (2.1), we have ||𝑧2β„Žξ…ž||≀𝑛(𝑧)+1βˆšπ‘›2+1(π‘§βˆˆπ•Œ).(2.5)
Hence, using Lemma 1.1, we have βˆ‘β„Ž(𝑧)βˆ—0(1,𝑛,0).
From (2.3), we can write π‘§β„Žξ…ž(𝑧)=1β„Ž(𝑧)ξ‚Έ(π‘βˆ’π›Ό)cosπœ†π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)ξ‚Ή.𝑓(𝑧)+(𝛼cosπœ†+𝑖𝑝sinπœ†)(2.6)
Since βˆ‘β„Ž(𝑧)βˆˆβˆ—0(1,𝑛,0), it implies that Re(βˆ’π‘§β„Žξ…ž(𝑧)/β„Ž(𝑧))>0. Therefore, we get 1ξ‚Έξ‚΅βˆ’(π‘βˆ’π›Ό)cosπœ†Reπ‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)ξ‚Άξ‚Ήξ‚΅βˆ’π‘“(𝑧)βˆ’π›Όcosπœ†=Reπ‘§β„Žξ…ž(𝑧)ξ‚Άβ„Ž(𝑧)>0(2.7) or ξ‚΅βˆ’Reπ‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)𝑓(𝑧)>𝛼cosπœ†,(2.8) and this implies that βˆ‘π‘“(𝑧)βˆˆβˆ—πœ†(𝑝,𝑛,𝛼).
If we take πœ†=0, we obtain the following result.

Corollary 2.2. If βˆ‘π‘“(𝑧)∈(𝑝,𝑛) satisfies ||||(𝑧𝑝𝑓(𝑧))1/(π‘βˆ’π›Ό)ξ‚»π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)ξ‚Ό+||||<1𝑓(𝑧)+𝛼(π‘βˆ’π›Ό)√2(π‘βˆ’π›Ό)(π‘§βˆˆπ•Œ),(2.9) then βˆ‘π‘“(𝑧)βˆˆβˆ—(𝑝,𝑛,𝛼).

Theorem 2.3. If βˆ‘π‘“(𝑧)∈(𝑝,𝑛) satisfies |||||𝑧𝑝+1π‘“ξ…ž(𝑧)ξ‚Άβˆ’π‘π‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†ξ‚»π‘’π‘–πœ†ξ‚΅π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Άξ‚Ό|||||<(𝑧)+1+𝛼cosπœ†+𝑖𝑝sinπœ†+(π‘βˆ’π›Ό)cosπœ†(𝑛+1)(π‘βˆ’π›Ό)cosπœ†βˆš(𝑛+1)2(+1π‘§βˆˆπ•Œ),(2.10) then βˆ‘π‘“(𝑧)βˆˆπœ†π‘(𝑝,𝑛,𝛼).

Proof. Let us set ξ€œβ„Ž(𝑧)=βˆ’π‘§01𝑑2ξ‚΅βˆ’π‘‘π‘+1π‘“ξ…ž(𝑑)π‘ξ‚Άπ‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†1𝑑𝑑=𝑧+π‘›βˆ’π‘+1π‘’π‘›π‘π‘–πœ†π‘Žπ‘›π‘§(π‘βˆ’π›Ό)cosπœ†π‘›+β‹―.(2.11)
Also let 𝑔(𝑧)=βˆ’π‘§β„Žξ…ž1(𝑧)=π‘§ξ‚΅βˆ’π‘§π‘+1π‘“ξ…ž(𝑧)π‘ξ‚Άπ‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†=1𝑧+π‘βˆ’π‘›βˆ’1π‘π‘’π‘–πœ†π‘Žπ‘›π‘§(π‘βˆ’π›Ό)cosπœ†π‘›+β‹―.(2.12)
Then clearly β„Ž(𝑧) and βˆ‘π‘”(𝑧)∈(1,𝑛). Now 1𝑔(𝑧)=π‘§ξ‚΅βˆ’π‘§π‘+1π‘“ξ…ž(𝑧)π‘ξ‚Άπ‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†.(2.13)
Differentiating logarithmically and then simple computation gives us ||𝑧2π‘”ξ…ž||=|||||𝑧(𝑧)+1𝑝+1π‘“ξ…ž(𝑧)ξ‚Άβˆ’π‘π‘’π‘–πœ†/(π‘βˆ’π›Ό)cosπœ†1𝑒(π‘βˆ’π›Ό)cosπœ†π‘–πœ†ξ‚΅π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Άξ‚Ό|||||<𝑛(𝑧)+1+𝛼cosπœ†+𝑖𝑝sinπœ†+1βˆšπ‘›2.+1(2.14)
Therefore, by using Lemma 1.1, we have 𝑔(𝑧)=βˆ’π‘§β„Žξ…ž(𝑧)βˆˆβˆ—ξ“0(1,𝑛,0)(2.15) which implies that βˆ‘β„Ž(𝑧)∈0𝑐(1,𝑛,0). Since 1+π‘§β„Žξ…žξ…ž(𝑧)β„Žξ…ž=𝑒(𝑧)π‘–πœ†ξ‚»(π‘βˆ’π›Ό)cosπœ†π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ό(𝑧)+(𝑝+1)βˆ’1,(2.16) therefore ξ‚΅Re1+π‘§β„Žξ…žξ…ž(𝑧)β„Žξ…žξ‚Ά=1(𝑧)𝑒(π‘βˆ’π›Ό)cosπœ†Reπ‘–πœ†ξ‚΅1+π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ά(𝑧)+π‘π‘’π‘–πœ†ξ‚Ό=1βˆ’(π‘βˆ’π›Ό)cosπœ†(ξ‚»π‘βˆ’π›Ό)cosπœ†Reπ‘’π‘–πœ†ξ‚΅1+π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Άξ‚Ό.(𝑧)+𝛼cosπœ†(2.17) Since βˆ‘β„Ž(𝑧)∈0𝑐(1,𝑛,0), so βˆ’1ξ‚»(π‘βˆ’π›Ό)cosπœ†Reπ‘’π‘–πœ†ξ‚΅1+π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Άξ‚Ό(𝑧)+𝛼cosπœ†>0,(2.18) or βˆ’Reπ‘’π‘–πœ†ξ‚΅1+π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ά(𝑧)>𝛼cosπœ†.(2.19)
It follows that βˆ‘π‘“(𝑧)βˆˆπœ†π‘(𝑝,𝑛,𝛼).

Theorem 2.4. If βˆ‘π‘“(𝑧)∈(𝑝,𝑛) satisfies 𝑒Reπ‘–πœ†π‘§π‘“ξ…ž(𝑧)𝛼𝑓(𝑧)ξ‚Άξ‚΅π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ά>𝑀(𝑧)βˆ’124𝐿+𝑁(π‘§βˆˆπ•Œ),(2.20)
then βˆ‘π‘“(𝑧)βˆˆβˆ—πœ†(𝑝,𝑛,𝛽), where 0≀𝛼≀1, 0≀𝛽<𝑝 and 𝑛𝐿=𝛼(π‘βˆ’π›½)2𝛽+(π›½βˆ’π‘)cos2πœ†cosπœ†,𝑀=𝛼(π›½βˆ’π‘)(1βˆ’π›½cosπœ†)sin2πœ†cosπœ†,𝑁=𝛼2cos2πœ†+sin2π‘›πœ†βˆ’2(π›½βˆ’π‘)cosπœ†+(1+𝛼)𝛽cosπœ†.(2.21)

Proof. Let us set π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)=[]𝑓(𝑧)(π›½βˆ’π‘)β„Ž(𝑧)βˆ’π›½cosπœ†βˆ’π‘–sinπœ†.(2.22)
Then β„Ž(𝑧) is analytic in π•Œ with 𝑝(0)=1.
Taking logarithmic differentiation of (2.22) and then by simple computation, we obtain π‘’π‘–πœ†π‘§π‘“ξ…ž(𝑧)𝛼𝑓(𝑧)π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Ά(𝑧)βˆ’1=π΄π‘§β„Žξ…ž(𝑧)+π΅β„Ž2ξ€·(𝑧)+πΆβ„Ž(𝑧)+𝐷=Ξ¨β„Ž(𝑧),π‘§β„Žξ…žξ€Έ(𝑧),𝑧(2.23) with 𝐴=(π›½βˆ’π‘)𝛼cosπœ†,𝐡=π›Όπ‘’βˆ’π‘–πœ†(π›½βˆ’π‘)2cos2ξ€Ίπœ†,𝐢=(π‘βˆ’π›½)π›Όπ‘’βˆ’π‘–πœ†ξ€·2𝛽cos2ξ€Έπœ†+𝑖sin2πœ†+(1+𝛼)π‘’βˆ’π‘–πœ†ξ€»,cosπœ†π·=π›Όπ‘’βˆ’π‘–πœ†ξ€·π›½2cos2πœ†βˆ’sin2ξ€Έπœ†+𝑖𝛽sin2πœ†+(1+𝛼)(𝛽cosπœ†βˆ’π‘–sinπœ†).(2.24)
Now for all real π‘₯ and 𝑦 satisfying π‘¦β‰€βˆ’(𝑛/2)(1+π‘₯2), we have Ξ¨(𝑖π‘₯,𝑦,𝑧)=π΄π‘¦βˆ’π΅π‘₯2+𝐢(𝑖π‘₯)+𝐷.(2.25)
Reputing the values of 𝐴, 𝐡, 𝐢, and 𝐷 and then taking real part, we obtain ReΞ¨(𝑖π‘₯,𝑦,𝑧)β‰€βˆ’πΏπ‘₯2ξƒ©βˆš+𝑀π‘₯+𝑁=βˆ’π‘€πΏπ‘₯βˆ’2√𝐿ξƒͺ2+𝑀2<𝑀4𝐿+𝑁24𝐿+𝑁,(2.26) where 𝐿, 𝑀, and 𝑁 are given in (2.21).
Let Ξ©={π‘€βˆΆRe𝑀>(𝑀2/4𝐿)+𝑁}. Then Ξ¨(β„Ž(𝑧),π‘§β„Žξ…ž(𝑧),𝑧)∈Ω and Ξ¨(𝑖π‘₯,𝑦,𝑧)βˆ‰Ξ©, for all real π‘₯ and 𝑦 satisfying π‘¦β‰€βˆ’(𝑛/2)(1+π‘₯2), π‘§βˆˆπ•Œ. By using Lemma 1.2, we have Reβ„Ž(𝑧)>0, that is βˆ‘π‘“(𝑧)βˆˆβˆ—πœ†(𝑝,𝑛,𝛽).
If we put πœ†=0, we obtain the following result.

Corollary 2.5. If βˆ‘π‘“(𝑧)∈(𝑝,𝑛) satisfies Reπ‘§π‘“ξ…ž(𝑧)𝛼𝑓(𝑧)π‘§π‘“ξ…žξ…ž(𝑧)π‘“ξ…žξ‚Άξ‚€π›½(𝑧)βˆ’1>𝛼2βˆ’π‘›2(π›½βˆ’π‘)+(1+𝛼)𝛽(π‘§βˆˆπ•Œ),(2.27) then βˆ‘π‘“(𝑧)βˆ—(𝑝,𝑛,𝛽), where 0≀𝛼≀1, 0≀𝛽<𝑝.

Theorem 2.6. For π‘—βˆˆ{1,β€¦π‘š}, let π‘“π‘—βˆ‘(𝑧)∈(𝑝,𝑛) and satisfy (2.9). If π‘šξ“π‘—=1𝛼𝑗<𝑝+1,π‘βˆ’π›½(2.28) then π»π‘š,π‘βˆ‘(𝑧)βˆˆπ‘(𝑝,𝑛,𝜁), where 𝜁>𝑝.

Proof. From (1.4), we obtain 𝑧𝑝+1π»ξ…žπ‘š,𝑝(𝑧)+(𝑝+1)π‘§π‘π»π‘š,𝑝(𝑧)=π‘šξ‘π‘—=1𝑧𝑝𝑓𝑗(𝑧)𝛼𝑗.(2.29)
Differentiating again logarithmically and then by simple computation, we get π‘§π»ξ…žξ…žπ‘š,𝑝(𝑧)π»ξ…žπ‘š,𝑝𝑝(𝑧)+1+2ξ€Έπ»βˆ’1π‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝𝐻(𝑧)+2𝑝=1+(𝑝+1)π‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝(𝑧)ξƒ­ξƒ¬π‘šξ“π‘—=1π›Όπ‘—ξƒ©π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒ­,(𝑧)+π‘βˆ’1(2.30) or, equivalently we can write βˆ’ξƒ©π‘§π»ξ…žξ…žπ‘š,𝑝(𝑧)π»ξ…žπ‘š,𝑝ξƒͺ=𝐻(𝑧)+1π‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝(𝑧)(𝑝+1)π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒͺ+𝑝(𝑧)βˆ’π‘+12ξ€Έξƒ­+βˆ’1π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗(ξƒͺ𝑧)βˆ’π‘+(1+2𝑝).(2.31) Now taking real part on both sides, we obtain ξƒ©βˆ’Reπ‘§π»ξ…žξ…žπ‘š,𝑝(𝑧)π»ξ…žπ‘š,𝑝ξƒͺ𝐻(𝑧)+1=Reπ‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝(𝑧)(𝑝+1)π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒͺ+𝑝(𝑧)βˆ’π‘+12ξ€Έξƒ­+βˆ’1π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’Reπ‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗(ξƒͺ𝑧)βˆ’π‘+(1+2𝑝).(2.32) This further implies that ξƒ©βˆ’Reπ‘§π»ξ…žξ…žπ‘š,𝑝(𝑧)π»ξ…žπ‘š,𝑝ξƒͺ≀|||||𝐻(𝑧)+1π‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝(𝑧)(𝑝+1)π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒͺ+𝑝(𝑧)βˆ’π‘+12ξ€Έξƒ­|||||+βˆ’1π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’Reπ‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺ(𝑧)βˆ’π‘+(1+2𝑝).(2.33)
Let |||||𝐻𝜁=π‘š,𝑝(𝑧)π‘§π»ξ…žπ‘š,𝑝(𝑧)(𝑝+1)π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’π‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒͺ+𝑝(𝑧)βˆ’π‘+12ξ€Έξƒ­|||||+ξƒ¬βˆ’1π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’Reπ‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒ­.(𝑧)βˆ’π‘+(1+2𝑝)(2.34) Clearly we have ξƒ¬πœ>π‘šξ“π‘—=1π›Όπ‘—ξƒ©βˆ’Reπ‘§π‘“ξ…žπ‘—(𝑧)𝑓𝑗ξƒͺξƒ­.(𝑧)βˆ’π‘+(1+2𝑝)(2.35) Then by using (2.28) and Corollary 2.2, we obtain 𝜁>π‘šξ“π‘—=1𝛼𝑗(π›½βˆ’π‘)+(1+2𝑝)>𝑝.(2.36) Therefore π»π‘š,π‘βˆ‘(𝑧)βˆˆπ‘(𝑝,𝑛,𝜁) with 𝜁>𝑝.
Making use of (2.27) and Corollary 2.5, one can prove the following result.

Theorem 2.7. For π‘—βˆˆ{1,β€¦π‘š}, let π‘“π‘—βˆ‘(𝑧)∈(𝑝,𝑛) and satisfy (2.27). If π‘šξ“π‘—=1𝛼𝑗<𝑝+1,π‘βˆ’π›½(2.37) then π»π‘š,π‘βˆ‘(𝑧)βˆˆπ‘(𝑝,𝑛,𝜁), where 𝜁>𝑝.


The author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.


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