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International Journal of Mathematical Archive- 7(1), Jan. – 2016 166
UNIQUENESS OF SOME GENERAL CLASS
OF DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS
HARINA P. WAGHAMORE*
1, HUSNA V.
21
Department of Mathematics,
Jnanabharathi Campus, Bangalore University, Bengaluru-560 056, India.
2
Department of Mathematics,
Jnanabharathi Campus, Bangalore University, Bengaluru-560 056, India.
(Received On: 11-01-16; Revised & Accepted On: 31-01-16)
ABSTRACT
W
e discuss some general class of differential polynomials of meromorphic functions and obtain two theorems whichimprove a result of Abhijit Banerjee [17].
2010 Mathematics Subject Classification: Primary 30D35.
Key words and phrases: Meromorphic functions, Sharing of values, Small function.
1. INTRODUCTION AND MAIN RESULTS
Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions, 𝑎𝑎 ∈ ℂ ∪{∞}. We say that 𝑓𝑓and 𝑔𝑔 share the value 𝑎𝑎 CM (counting multiplicities) if 𝑓𝑓 − 𝑎𝑎 and 𝑔𝑔 − 𝑎𝑎 have the same zeros with the same multiplicities and if we do not consider the multiplicities, then 𝑓𝑓and 𝑔𝑔 are said to share the value a IM (ignoring multiplicities). We denote by 𝑇𝑇(𝑟𝑟) the maximum of 𝑇𝑇(𝑟𝑟,𝑓𝑓) and 𝑇𝑇(𝑟𝑟,𝑔𝑔). The notation S(r) denotes any quantity satisfying 𝑆𝑆(𝑟𝑟) = 0(𝑇𝑇(𝑟𝑟)) as 𝑟𝑟 → ∞, outside a set of finite linear measure.
We use 𝐼𝐼 to denote any set of infinite linear measure of 0 < 𝑟𝑟 < ∞.
Due to Nevanlinna [9], it is well known that if 𝑓𝑓 and 𝑔𝑔 share four distinct values CM, then 𝑓𝑓 is a Mobius transformation of 𝑔𝑔.
Yang and Hua showed that similar conclusions hold for certain types of differential polynomials when they share only one value. They proved the following result.
Theorem 1.1[12]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions, 𝑛𝑛 ≥11 an integer, and 𝑎𝑎 ∈ ℂ −{0}. If
𝑓𝑓𝑛𝑛𝑓𝑓′ and 𝑔𝑔𝑛𝑛𝑔𝑔′ share the value 𝑎𝑎 CM, then either 𝑓𝑓=𝑑𝑑𝑔𝑔 for some (n+1)th root of unity d or 𝑔𝑔(𝑧𝑧) =𝑐𝑐
1𝑒𝑒𝑐𝑐𝑧𝑧 and 𝑓𝑓(𝑧𝑧) =𝑐𝑐2𝑒𝑒−𝑐𝑐𝑧𝑧 where 𝑐𝑐,𝑐𝑐1 and 𝑐𝑐2 are constants satisfying (𝑐𝑐1𝑐𝑐2)𝑛𝑛+1𝑐𝑐2=−𝑎𝑎2.
Corresponding to entire functions, Xu and Qu proved the following result.
Theorem 1.2[10]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant entire functions, 𝑛𝑛 ≥12 an integer, and 𝑎𝑎 ∈ ℂ −{0}. If 𝑓𝑓𝑛𝑛𝑓𝑓′
and 𝑔𝑔𝑛𝑛𝑔𝑔′share the value 𝑎𝑎 IM, then either 𝑓𝑓=𝑑𝑑𝑔𝑔 for some (n+1)th root of unity d or 𝑔𝑔(𝑧𝑧) =𝑐𝑐1𝑒𝑒𝑐𝑐𝑧𝑧and
𝑓𝑓(𝑧𝑧) =𝑐𝑐2𝑒𝑒−𝑐𝑐𝑧𝑧 where 𝑐𝑐,𝑐𝑐1and 𝑐𝑐2 are constants and satisfy (𝑐𝑐1𝑐𝑐2)𝑛𝑛+1𝑐𝑐2=−𝑎𝑎2.
To state the next result, we require the following definition.
Definition 1.3[4, 5]: Let 𝑘𝑘 be a nonnegative integer or infinity. For 𝑎𝑎 ∈ ℂ −{∞}, denote by 𝐸𝐸𝑘𝑘(𝑎𝑎;𝑓𝑓) the set of all 𝑎𝑎 -points of 𝑓𝑓, where an 𝑎𝑎-point of multiplicity 𝑚𝑚 is counted 𝑚𝑚 times if 𝑚𝑚 ≤ 𝑘𝑘 and 𝑘𝑘 + 1 times if 𝑚𝑚 > 𝑘𝑘. If
𝐸𝐸𝑘𝑘(𝑎𝑎,𝑓𝑓) =𝐸𝐸𝑘𝑘(𝑎𝑎,𝑔𝑔) say that 𝑓𝑓,𝑔𝑔 share the value 𝑎𝑎 with weight 𝑘𝑘.
Corresponding Author: Harina P. Waghamore*
1,
1Department of Mathematics,
The definition implies that if 𝑓𝑓,𝑔𝑔 share a value a with weight 𝑘𝑘 then 𝑧𝑧0 is an 𝑎𝑎-point of 𝑓𝑓 with multiplicity 𝑚𝑚(≤ 𝑘𝑘) if and only it is an 𝑎𝑎-point of 𝑔𝑔 with multiplicity 𝑚𝑚(≤ 𝑘𝑘) and 𝑧𝑧0is an 𝑎𝑎-point of 𝑓𝑓 with multiplicity 𝑚𝑚(>𝑘𝑘) if and only if is an 𝑎𝑎-point of 𝑔𝑔 with multiplicity 𝑛𝑛(>𝑘𝑘) where 𝑚𝑚 is not necessarily equal to 𝑛𝑛.
We write 𝑓𝑓,𝑔𝑔 share (𝑎𝑎,𝑘𝑘) to mean that 𝑓𝑓; 𝑔𝑔 share the value 𝑎𝑎 with weight 𝑘𝑘. Since 𝐸𝐸𝑘𝑘(𝑎𝑎; 𝑓𝑓) = 𝐸𝐸𝑘𝑘(𝑎𝑎; 𝑔𝑔) implies
𝐸𝐸𝑝𝑝(𝑎𝑎; 𝑓𝑓) = 𝐸𝐸𝑝𝑝(𝑎𝑎; 𝑔𝑔) for any integer 𝑝𝑝(0≤ 𝑝𝑝<𝑘𝑘), clearly if 𝑓𝑓,𝑔𝑔 share (𝑎𝑎,𝑘𝑘), then 𝑓𝑓,𝑔𝑔 share (𝑎𝑎,𝑝𝑝) for any integer
𝑝𝑝(0≤ 𝑝𝑝<𝑘𝑘), Also we note that 𝑓𝑓,𝑔𝑔 share a value 𝑎𝑎 IM or CM if and only if 𝑓𝑓,𝑔𝑔 share (𝑎𝑎, 0) or (𝑎𝑎,∞), respectively.
Theorem 1.4[5]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions, 𝑛𝑛 ≥11 an integer, and 𝑎𝑎 ∈ ℂ −{0}. If 𝑓𝑓𝑛𝑛𝑓𝑓′ and 𝑔𝑔𝑛𝑛𝑔𝑔′ share (𝑎𝑎, 2), then either 𝑓𝑓=𝑑𝑑𝑔𝑔 for some (n+1)th root of unity 𝑑𝑑 or 𝑔𝑔(𝑧𝑧) =𝑐𝑐1𝑒𝑒𝑐𝑐𝑧𝑧 and 𝑓𝑓(𝑧𝑧) =𝑐𝑐2𝑒𝑒−𝑐𝑐𝑧𝑧 where
𝑐𝑐,𝑐𝑐1and 𝑐𝑐2 are constants and satisfy (𝑐𝑐1𝑐𝑐2)𝑛𝑛+1𝑐𝑐2=−𝑎𝑎2.
Theorem 1.5[17]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions such that 𝑛𝑛 > 22−[5Θ(∞; 𝑓𝑓) +
5Θ(∞; 𝑔𝑔) + min{Θ(∞; 𝑓𝑓),Θ(∞; 𝑔𝑔)}], where 𝑛𝑛 is an integer. If for 𝑎𝑎 ∈ ℂ −{0}, 𝑓𝑓𝑛𝑛𝑓𝑓′ and 𝑔𝑔𝑛𝑛𝑔𝑔′ share (𝑎𝑎, 0), then either 𝑓𝑓 = 𝑑𝑑𝑔𝑔 for some (n+1)th root of unity 𝑑𝑑 or 𝑔𝑔(𝑧𝑧) =𝑐𝑐1𝑒𝑒𝑐𝑐𝑧𝑧 and 𝑓𝑓(𝑧𝑧) =𝑐𝑐2𝑒𝑒−𝑐𝑐𝑧𝑧 where 𝑐𝑐,𝑐𝑐1and 𝑐𝑐2are constants and satisfy (𝑐𝑐1𝑐𝑐2)𝑛𝑛+1𝑐𝑐2=−𝑎𝑎2.
Theorem 1.6[17]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions and 𝑛𝑛> max�8, 12−{3𝛩𝛩(∞,𝑓𝑓) + 3𝛩𝛩(∞,𝑔𝑔)}� an integer. If for 𝑎𝑎 ∈ ℂ −{0}, 𝑓𝑓𝑛𝑛𝑓𝑓′ and 𝑔𝑔𝑛𝑛𝑔𝑔′ share (𝑎𝑎, 1), then either 𝑓𝑓 = 𝑑𝑑𝑔𝑔 for some (n+1)th root of unity 𝑑𝑑 or
𝑔𝑔(𝑧𝑧) =𝑐𝑐1𝑒𝑒𝑐𝑐𝑧𝑧and 𝑓𝑓(𝑧𝑧) =𝑐𝑐2𝑒𝑒−𝑐𝑐𝑧𝑧 where 𝑐𝑐,𝑐𝑐1and 𝑐𝑐2are constants and satisfy (𝑐𝑐1𝑐𝑐2)𝑛𝑛+1𝑐𝑐2=−𝑎𝑎2.
Here in this paper, we partially extend Theorem 1.5 and Theorem 1.6 to a more general class of differential polynomials as.
Theorem 1.7: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions such that 𝑛𝑛 > 𝑘𝑘𝑚𝑚 + 3𝑚𝑚 + 4𝑘𝑘 + 14, where
𝑛𝑛,𝑚𝑚 and 𝑘𝑘 be positive integers. If for 𝑎𝑎 ∈ ℂ −{0}, 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)share (𝑎𝑎, 0), then either
𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)≡ 𝑎𝑎2 𝑜𝑜𝑟𝑟 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘) ≡ 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘).
Theorem 1.8: Let𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions such that 𝑛𝑛 > 𝑘𝑘𝑚𝑚 + 3𝑚𝑚 + 3𝑘𝑘 + 11, where
𝑛𝑛,𝑚𝑚 and 𝑘𝑘 be positive integers. If for 𝑎𝑎 ∈ ℂ −{0}, 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)share (𝑎𝑎, 1), then either
𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)≡ 𝑎𝑎2 𝑜𝑜𝑟𝑟 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)≡ 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘).
Definition 1.9[3]: For 𝑎𝑎 ∈ ℂ −{∞}, denote by 𝑁𝑁(𝑟𝑟,𝑎𝑎; 𝑓𝑓| = 1) the counting function of simple 𝑎𝑎-points of 𝑓𝑓. For a positive integer 𝑚𝑚, denote by 𝑁𝑁(𝑟𝑟,𝑎𝑎; 𝑓𝑓|≤ 𝑚𝑚)(𝑁𝑁(𝑟𝑟,𝑎𝑎; 𝑓𝑓|≥ 𝑚𝑚)) the counting function of those 𝑎𝑎-points of 𝑓𝑓 whose multiplicities are not greater(less) than 𝑚𝑚 where each 𝑎𝑎-point is counted according to its multiplicity.
𝑁𝑁�(𝑟𝑟,𝑎𝑎; 𝑓𝑓|≤ 𝑚𝑚)(𝑁𝑁��(𝑟𝑟,𝑎𝑎; 𝑓𝑓| ≥ 𝑚𝑚)� are defined similarly, where in counting the 𝑎𝑎-points of 𝑓𝑓 we ignore the multiplicities.
Also 𝑁𝑁(𝑟𝑟,𝑎𝑎; 𝑓𝑓| < 𝑚𝑚), (𝑁𝑁(𝑟𝑟,𝑎𝑎; 𝑓𝑓| > 𝑚𝑚))𝑁𝑁�(𝑟𝑟,𝑎𝑎; 𝑓𝑓| < 𝑚𝑚) and 𝑁𝑁�(𝑟𝑟,𝑎𝑎; 𝑓𝑓| > 𝑚𝑚) are defined analogously.
Definition 1.10[5]: Denote by 𝑁𝑁2(𝑟𝑟; 𝑎𝑎; 𝑓𝑓) the sum 𝑁𝑁�(𝑟𝑟,𝑎𝑎; 𝑓𝑓) + 𝑁𝑁�(𝑟𝑟,𝑎𝑎; 𝑓𝑓|≥2).
Definition 1.11[1, 15, 16]: Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions such that 𝑓𝑓 and 𝑔𝑔 share the value 1 IM. Let 𝑧𝑧0 be a 1-point of 𝑓𝑓 with multiplicity 𝑝𝑝, a 1-point of 𝑔𝑔 with multiplicity 𝑞𝑞. Denote by 𝑁𝑁���𝐿𝐿(𝑟𝑟, 1;𝑓𝑓) the counting function of those 1-points of 𝑓𝑓 and 𝑔𝑔 where𝑝𝑝> 𝑞𝑞, denote by 𝑁𝑁𝐸𝐸1)(𝑟𝑟, 1;𝑓𝑓) the counting function of those 1-points of
𝑓𝑓 and 𝑔𝑔 where 𝑝𝑝=𝑞𝑞= 1, and denote by 𝑁𝑁𝐸𝐸2)(𝑟𝑟, 1;𝑓𝑓) the counting function of those 1-points of 𝑓𝑓 and 𝑔𝑔 where
𝑝𝑝=𝑞𝑞 ≥ 2, each point in these counting functions is counted only once. In the same way, one can define
𝑁𝑁𝐿𝐿
���(𝑟𝑟, 1;𝑔𝑔),𝑁𝑁𝐸𝐸1)(𝑟𝑟, 1;𝑔𝑔),𝑁𝑁𝐸𝐸2)(𝑟𝑟, 1;𝑔𝑔).
Definition 1.12(cf. [1]): Let 𝑘𝑘 be a positive integer. Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions such that 𝑓𝑓 and 𝑔𝑔 share the value 1 IM. Let 𝑧𝑧0 be a 1-point of 𝑓𝑓 with multiplicity 𝑝𝑝, and a 1-point of 𝑔𝑔 with multiplicity 𝑞𝑞. Denote by 𝑁𝑁�𝑓𝑓>𝑘𝑘(𝑟𝑟, 1;𝑔𝑔) the reduced counting function of those 1-points of 𝑓𝑓 and 𝑔𝑔 such that 𝑝𝑝>𝑞𝑞=𝑘𝑘.𝑁𝑁�𝑔𝑔>𝑘𝑘(𝑟𝑟, 1;𝑓𝑓) is defined analogously.
Definition 1.13([4, 5]): Let 𝑓𝑓,𝑔𝑔 share a value IM. Denote by 𝑁𝑁���∗(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) the reduced counting function of those 𝑎𝑎 -points of 𝑓𝑓 whose multiplicities differ from the multiplicities of the corresponding 𝑎𝑎-points of 𝑔𝑔.
© 2016, IJMA. All Rights Reserved 168 Definition 1.14([6]): Let 𝑎𝑎,𝑏𝑏 ∈ ℂ ∪{∞}. Denote by 𝑁𝑁(𝑟𝑟,𝑎𝑎;𝑓𝑓|𝑔𝑔=𝑏𝑏) the counting function of those 𝑎𝑎-points of 𝑓𝑓, counted according to multiplicity, which are the 𝑏𝑏-points of 𝑔𝑔.
Definition 1.15([6]): Let 𝑎𝑎,𝑏𝑏 ∈ ℂ ∪{∞}. Denote by 𝑁𝑁(𝑟𝑟,𝑎𝑎;𝑓𝑓|𝑔𝑔 ≠ 𝑏𝑏) the counting function of those 𝑎𝑎-points of
𝑓𝑓, counted according to multiplicity, which are not the 𝑏𝑏-points of 𝑔𝑔.
2. LEMMAS
In this section, we present some lemmas which will be needed in the sequel. Let 𝑓𝑓,𝑔𝑔,𝐹𝐹,𝐺𝐺 be four nonconstant meromorphic functions. Henceforth, We will denote by h and H the following two functions:
ℎ=�𝑓𝑓𝑓𝑓′ −′′ 𝑓𝑓 −2𝑓𝑓′1� − �𝑔𝑔𝑔𝑔′′′ −𝑔𝑔 −2𝑔𝑔′1�,𝐻𝐻=�𝐹𝐹𝐹𝐹′ −′′ 𝐹𝐹 −2𝐹𝐹′1� − �𝐺𝐺𝐺𝐺′′′ −𝐺𝐺 −2𝐺𝐺′1�,
Lemma 2.1[15]: Let 𝑓𝑓,𝑔𝑔 be two nonconstant meromorphic functions such that they share (1, 0) and ℎ ≠ 0. Then
𝑇𝑇(𝑟𝑟,𝑓𝑓)≤ 𝑁𝑁2�𝑟𝑟,1𝑓𝑓�+𝑁𝑁2(𝑟𝑟,𝑓𝑓) +𝑁𝑁2�𝑟𝑟,1𝑔𝑔�+𝑁𝑁2(𝑟𝑟,𝑔𝑔) + 2𝑁𝑁� �𝑟𝑟,1𝑓𝑓�+ 2𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑁𝑁� �𝑟𝑟,𝑔𝑔�1 +𝑁𝑁�(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
Lemma 2.2[17]: Let 𝑓𝑓,𝑔𝑔 be two nonconstant meromorphic functions such that they share (1,1) and ℎ ≠ 0. Then
𝑇𝑇(𝑟𝑟,𝑓𝑓)≤ 𝑁𝑁2�𝑟𝑟,1𝑓𝑓�+𝑁𝑁2(𝑟𝑟,𝑓𝑓) +𝑁𝑁2�𝑟𝑟,𝑔𝑔�1 +𝑁𝑁2(𝑟𝑟,𝑔𝑔) +12𝑁𝑁� �𝑟𝑟,𝑓𝑓�1 +12𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
Lemma 2.3[18]: Let 𝑓𝑓 and 𝑔𝑔 be two meromorphic functions. If 𝑓𝑓 and 𝑔𝑔 share 1 CM, then one of the following must occur:
(i) 𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)≤2�𝑁𝑁2�𝑟𝑟,1
𝑓𝑓�+𝑁𝑁2�𝑟𝑟,
1
𝑔𝑔�+𝑁𝑁2(𝑟𝑟,𝑓𝑓) +𝑁𝑁2(𝑟𝑟,𝑔𝑔)�+𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
(ii) either 𝑓𝑓 ≡ 𝑔𝑔 or 𝑓𝑓𝑔𝑔 ≡1.
Lemma 2.4[15]: Let 𝑓𝑓 be a nonconstant meromorphic function. Then
𝑁𝑁 �𝑟𝑟, 1
𝑓𝑓(𝑘𝑘)� ≤ 𝑘𝑘𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑁𝑁 �𝑟𝑟, 1
𝑓𝑓�+𝑆𝑆(𝑟𝑟,𝑓𝑓).
3. PROOF OF THEOREM 1.7
Let 𝐹𝐹=𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)
𝑎𝑎 and 𝐺𝐺 =
𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)
𝑎𝑎 . Since 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘) share (𝑎𝑎, 0). If possible, we suppose that
𝐻𝐻 ≢0. Then by Lemma 2.1, we obtain
𝑇𝑇(𝑟𝑟,𝐹𝐹)≤ 𝑁𝑁2�𝑟𝑟,1𝐹𝐹�+𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2�𝑟𝑟,1𝐺𝐺�+𝑁𝑁2(𝑟𝑟,𝐺𝐺) + 2𝑁𝑁� �𝑟𝑟,𝐹𝐹�1 + 2𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁� �𝑟𝑟,1𝐺𝐺�+𝑁𝑁�(𝑟𝑟,𝐺𝐺) +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺)
We see that
𝑁𝑁2�𝑟𝑟,𝐹𝐹�1 +𝑁𝑁2(𝑟𝑟,𝐹𝐹)≤ 𝑁𝑁2�𝑟𝑟,𝑓𝑓𝑛𝑛(𝑓𝑓1𝑚𝑚)(𝑘𝑘)�+𝑁𝑁2�𝑟𝑟,𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)�+𝑆𝑆(𝑟𝑟,𝑓𝑓)
≤ 𝑁𝑁2�𝑟𝑟,𝑓𝑓1𝑛𝑛�+𝑁𝑁2�𝑟𝑟,(𝑓𝑓𝑚𝑚1)(𝑘𝑘)�+ 2𝑁𝑁��𝑟𝑟,𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)�+𝑆𝑆(𝑟𝑟,𝑓𝑓)
≤2𝑁𝑁� �𝑟𝑟,1𝑓𝑓�+𝑁𝑁 �𝑟𝑟,(𝑓𝑓𝑚𝑚1)(𝑘𝑘)�+ 2𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓)
≤2𝑁𝑁� �𝑟𝑟,1𝑓𝑓�+𝑚𝑚𝑁𝑁 �𝑟𝑟,1𝑓𝑓�+ (𝑘𝑘+ 2)𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓) = (𝑘𝑘+𝑚𝑚+ 4)𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓).
Therefore,
𝑁𝑁2�𝑟𝑟,1𝐹𝐹�+𝑁𝑁2(𝑟𝑟,𝐹𝐹)≤(𝑘𝑘+𝑚𝑚+ 4)𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓) (3.1)
On the similar lines we can write (3.1) for the function G as
𝑁𝑁2�𝑟𝑟,1𝐺𝐺�+𝑁𝑁2(𝑟𝑟,𝐺𝐺)≤(𝑘𝑘+𝑚𝑚+ 4)𝑇𝑇(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑔𝑔) (3.2)
Since
𝑛𝑛𝑇𝑇(𝑟𝑟,𝑓𝑓) =𝑇𝑇(𝑟𝑟,𝑓𝑓𝑛𝑛) =𝑇𝑇 �𝑟𝑟,𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)𝑎𝑎
≤ 𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑇𝑇 �𝑟𝑟, 1
(𝑓𝑓𝑚𝑚)(𝑘𝑘)�+𝑇𝑇(𝑟𝑟,𝑎𝑎) +𝑆𝑆(𝑟𝑟,𝑓𝑓)
=𝑇𝑇(𝑟𝑟,𝐹𝐹) + (𝑘𝑘𝑚𝑚+𝑚𝑚)𝑇𝑇 �𝑟𝑟,𝑓𝑓1�+𝑆𝑆(𝑟𝑟,𝑓𝑓) (3.3) Therefore
(𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 𝑚𝑚)𝑇𝑇(𝑟𝑟,𝑓𝑓)≤ 𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝑓𝑓) (3.4)
Similarly
(𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 𝑚𝑚)𝑇𝑇(𝑟𝑟,𝑔𝑔)≤ 𝑇𝑇(𝑟𝑟,𝐺𝐺) +𝑆𝑆(𝑟𝑟,𝑔𝑔) (3.5)
(𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 𝑚𝑚){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)}≤{𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑇𝑇(𝑟𝑟,𝐺𝐺)} +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔) (3.6)
Suppose that
𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑇𝑇(𝑟𝑟,𝐺𝐺)≤2�𝑁𝑁2�𝑟𝑟,1𝐹𝐹�+𝑁𝑁2(𝑟𝑟,𝐹𝐹)�+ 2{𝑁𝑁2�𝑟𝑟,𝐺𝐺�1 +𝑁𝑁2(𝑟𝑟,𝐺𝐺)} + 3�𝑁𝑁� �𝑟𝑟,1𝐹𝐹�+𝑁𝑁� �𝑟𝑟,𝐺𝐺��1 + 3{𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁�(𝑟𝑟,𝐺𝐺)} +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔).
Then from (3.1), (3.2), (3.6) and (3.7) we have
(𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 𝑚𝑚){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)}
≤2{𝑁𝑁2�𝑟𝑟,𝐹𝐹�1 +𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2�𝑟𝑟,𝐺𝐺�1 +𝑁𝑁2(𝑟𝑟,𝐺𝐺)} + 3�𝑁𝑁 �𝑟𝑟,𝐹𝐹�1 +𝑁𝑁 �𝑟𝑟,𝐺𝐺�1 +𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁�(𝑟𝑟,𝐺𝐺)� +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
≤2(𝑘𝑘+𝑚𝑚+ 4){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)} + 3�𝑁𝑁 �𝑟𝑟,1
𝑓𝑓�+𝑁𝑁 �𝑟𝑟, 1
𝑔𝑔�+ (𝑘𝑘+ 1)�𝑁𝑁� �𝑟𝑟, 1
𝑓𝑓�+𝑁𝑁� �𝑟𝑟, 1
𝑔𝑔��+𝑘𝑘{𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑁𝑁�(𝑟𝑟,𝑔𝑔)} + 2𝑁𝑁�(𝑟𝑟,𝑓𝑓) + 2𝑁𝑁�(𝑟𝑟,𝑔𝑔)�+𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
≤{4𝑘𝑘+ 2𝑚𝑚+ 14}{𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)} +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
which implies that
{𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 3𝑚𝑚 − 4𝑘𝑘 −14}{𝑇𝑇(𝑟𝑟,𝑓𝑓) + 𝑇𝑇(𝑟𝑟,𝑔𝑔)}≤ 𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
a contradiction since 𝑛𝑛>𝑘𝑘𝑚𝑚+ 3𝑚𝑚+ 4𝑘𝑘+ 14, where 𝑚𝑚>𝑘𝑘 −1.
Then by Lemma 2.3, it follows that either
𝐹𝐹.𝐺𝐺 ≡1 or 𝐹𝐹 ≡ 𝐺𝐺 that is either 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)≡ 𝑎𝑎2 or 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)=𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘).
4. PROOF OF THEOREM 1.8
Let 𝐹𝐹=𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)
𝑎𝑎 and 𝐺𝐺=
𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)
𝑎𝑎 . Since 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘) share (𝑎𝑎, 1). Suppose that 𝐻𝐻 ≢0. Then by
Lemma 2.2, we obtain
𝑇𝑇(𝑟𝑟,𝐹𝐹)≤ 𝑁𝑁2�𝑟𝑟,𝐹𝐹1�+𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2�𝑟𝑟,𝐺𝐺1�+𝑁𝑁2(𝑟𝑟,𝐺𝐺) +12𝑁𝑁� �𝑟𝑟,𝐹𝐹1�+12𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺)
Suppose that
𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑇𝑇(𝑟𝑟,𝐺𝐺)≤2{𝑁𝑁2�𝑟𝑟,𝐹𝐹1�+𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2�𝑟𝑟,𝐺𝐺1�+𝑁𝑁2(𝑟𝑟,𝐺𝐺)} +12 {���� �𝑟𝑟,𝑁𝑁 𝐹𝐹1�+𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁� �𝑟𝑟,𝐺𝐺1� +𝑁𝑁�(𝑟𝑟,𝐺𝐺)} +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺) (4.1)
Then from (3.1),(3.2),(3.6) and (4.1) we have
(𝑛𝑛 − 𝑘𝑘𝑚𝑚 − 𝑚𝑚){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)}
≤2{𝑁𝑁2�𝑟𝑟,𝐹𝐹1�+𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2�𝑟𝑟,𝐺𝐺1�+𝑁𝑁2(𝑟𝑟,𝐺𝐺)} +12 {���� �𝑟𝑟,𝑁𝑁 𝐹𝐹1�+𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁� �𝑟𝑟,𝐺𝐺1�+𝑁𝑁�(𝑟𝑟,𝐺𝐺)} +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺)
≤2(𝑘𝑘+𝑚𝑚+ 4){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)}
+12�𝑁𝑁 �𝑟𝑟,1𝑓𝑓�+ (𝑘𝑘+ 1)𝑁𝑁� �𝑟𝑟,𝑓𝑓�1 +𝑘𝑘𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑁𝑁 �𝑟𝑟,1𝑓𝑓�+ (𝑘𝑘+ 1)𝑁𝑁� �𝑟𝑟,𝑔𝑔�1 +𝑘𝑘𝑁𝑁�(𝑟𝑟,𝑔𝑔) + 2𝑁𝑁�(𝑟𝑟,𝑓𝑓)
+ 2𝑁𝑁�(𝑟𝑟,𝑔𝑔)�+𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔) ≤(3𝑘𝑘+ 2𝑚𝑚+ 11){𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)} +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)
which implies that
© 2016, IJMA. All Rights Reserved 170 Then by Lemma 2.3, it follows that either
𝐹𝐹.𝐺𝐺 ≡1 or 𝐹𝐹 ≡ 𝐺𝐺 that is either 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘)≡ 𝑎𝑎2 or 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚)(𝑘𝑘)=𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚)(𝑘𝑘).
ACKNOWLEDGEMENT
The author (HV) is grateful to the University Grants Commission (UGC), New Delhi, India for supporting her research work by providing her with a Maulana Azad National Fellowship (MANF).
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Source of support: University Grants Commission (UGC), New Delhi, India. Conflict of interest: None Declared
