ON MEROMORPHIC FUNCTION WITH MAXIMAL DEFICIENCY SUM AND IT'S DIFFERENCE OPERATORS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 1, P. 121-135

YAK 517.547

DOI 10.46698/g4967-8526-0651-y

ON MEROMORPHIC FUNCTION WITH MAXIMAL DEFICIENCY SUM AND IT'S DIFFERENCE OPERATORS

A. Shaw1

1 Balagarh High School, Balagarh, Hooghly, West Bengal 712501, India E-mail: ashaw2912@gmail.com

Abstract. The paper deals with characteristic funtion and deficiency of a meromorphic function. We

mainly focused on the relation between the characteristic function of a product of difference operators

with the characteristic function of a meromorphic function with maximal deficiency sum. The concept of

maximal deficiency sum of a meromorphic function is employed as an effective tool for our research. In the

same context, the notion of a difference polynomial of a difference operator is discussed. The paper contains

the details analysis and discussion of some asymptotic behaviour of the product of difference operators,

such as lim T(r'n-=' Ai«f) lim ^^Li a„,/) jV(r,TC;nLi A^/)+Ar(r,o;nLi ^J) ptc and

sucn as Hindoo T(r /) , iim^oo T(r-,n|=1 a^/) > umr->-°° T(r,n|=1A^/) etc. ana

same resolution and discussion also developed for the difference polynomial of difference operators. Several

innovative idea to establish some inequalities on the zeros and poles for J}?=1 Ani f and L(An f) are also

introduced. We broadly elaborate our results with many remarks and corollaries, and give two excellent

examples for proper justification of our results. The results on product and polynomial of difference

operators of our article improved and generalised the results of Z. Wu.

Key words: transcendental meromorphic function, deficiency sum, difference operators, product of difference operators.

AMS Subject Classification: 30D35, 39A05.

For citation: Shaw, A. On Meromorphic Function with Maximal Deficiency Sum and it's Difference Operators, Vladikavkaz Math. J., 2022, vol. 24, no. 1, pp. 121-135. DOI: 10.46698/g4967-8526-0651-y.

1. Introduction, Definations and Notations

Let f be a meromorphic function defined in C and a € C. We adopt some notations from the NNevanlinna theory of meromorphic functions, such as m(r, f), N(r, to; f), m(r, a) = m(r, jz^), N(r, 0; /) etc., for details see [1-3]. We defined T(r, /) = N(r, oo; /) + m(r, f) and called Nevanlinna's characteristic function, and defined the quantity S(r, f) such that S(r, f) = o(T(r, f)) as r — to, outside of a set of finite measure.

Definition 1.1. Deficiency of a € C with respect to a meromorphic function f is denoted by 5(a, f) and defined as

By the Nevanlinna's second fundamental theorem, it can be easily shown that

E 5(a,f) < 2.

aeCU{^}

© 2022 Shaw, A.

If equality holds in the above relation, we say that f is of maximal deficiency sum. Definition 1.2. The order of a meromorphic function f is denoted by a(f ) and define

by

m r log T(r,f) ait) = lim sup—:-.

r^œ log r

Let P(z, f ) be rational function in f with small meromorphic coefficients. Then by Valiron-Mo'honko identity we have

T (r, P (z, f )) = degf (p)T (r, f ) + S (r, f ). (1.1)

Hulburd and Korhonen [4], define difference operator by Anf (z) = f (z + n) — f (z), where n € C \ {0}. We take product of the difference operators by H¿=1 Anif, where n G C \ {0}, i = 1, 2,... ,q(q G Z+ ).

Definition 1.3. The q-th order difference operator A^f(z) is defined by A^f(z) = A^-1(Anf (z)), where q (^ 2) g N and n g C\{0}, while the difference polynomial of difference operator is given by L(Anf) = ^i=1 aiA\1f, where ai (i = 1,2,..., q) are nonzero constants. We can also deduce that,

q /

q

An f = £ (T)f (z + (q - On).

i=0 w

A. Edrei [5] and A. Weitsman [6] proved independently the following results:

Theorem A. Let f (z) be a transcendental meromorphic function with maximal deficiency sum. Then,

r—^^o T (r, f ) In 2000, Fang [7] obtained the following result:

Theorem B. Let f (z) be a transcendental meromorphic function with maximal deficiency sum with finite order. Then

2(1 — S(oo, /)) K(f)= 2-6(oo,f) '

where

= JV(r,oo;/') + iy(nO;/')

v 7 r—^ T(r,f')

In 2013, Z. Wu [8] proved the following results:

Theorem C. Let f (z) be a transcendental meromorphic function with maximal deficiency sum with the order less than one. Then,

V = 2-¿(00,/);

r—^ T (r, f )

JV(r,0; A,/) _

Theorem D. Let f (z) be a transcendental meromorphic function with maximal deficiency sum with the order less than one. Then,

2(1 — 5(oo, /)) 0 vf) ^ 2-¿(oo,/) '

where

K (An f) = lim

N(r,^]Avf)+N(r,0]Avf) ™ T(r, A„/)

Theorem E. Let f(z) be a transcendental meromorphic function with order is less than one and ¿(to, f) = 1. Then,

£ ¿(of) < ¿(0, Anf).

ae€U{^}

In the paper, we consider the product of difference operators and difference polynomial of difference operators of transcendental meromorphic functions with maximal deficiency sum and established some results generalizing the results of Z. Wu [8].

2. Main Results

In this section, we present our main results:

Theorem 2.1. Let f(z) be a transcendental meromorphic function with maximal deficiency sum of order less than one. Then

T(r, ft Ani/) v i=i 7

0 1i£n -t/v f\-=q(2-5(oo,f));

T (r, / )

^r, 0; = Am f

ii) lim -^- = 0.

r—ro / X . \

T(r, n Anif v i=i 7

Corollary 2.1. The deficiency of 0 with respect to Hq=1 Anif is / q x ^r, 0; ft Ani f)

¿ 0, []\J = 1 " Urn r1 , = 1-

V ^=1 7 T(r,n Ani f)

v i=i 7

Remark 2.1. If we put q = 1, then Theorem 2.1 coincide with Theorem C. Theorem 2.2. Let f(z) be a transcendental meromorphic function with maximal

deficiency sum of order less than one. Then,

q

-i=1 where

lim

vi=i 7

(2 - ¿(to,/)) '

^(r, to; ft Ani/) + ^(r, 0; ft Ani/

v ,-—1 / v „-—i

T(r^ Ani /) v i=i 7

Corollary 2.2. If in the above result if we take ¿(to, f) = 1, then we get

N(r, to; ft A^if)

lim --= 0.

™ ^(vk/)

i=1

q

Moreover, the deficiency of to with respect to Hq=1 Ani f is

^(r, to;E[ A^i/)

*( IT A»7i/) = l-rlim r1 = 1.

i=i 7 AVi /)

v i=i 7

Remark 2.2. If we put q = 1, then Theorem 2.2 coincide with Theorem D. Example 2.1. Take f(z) = where r is a complex constant. Observe that

lh II

(eni ez — t )(ez — t )

Then q=1 An,f # 0, ¿(0,f) = 1, ¿(—T,f) = 1, ¿(to, f) = 0. Thus f (z) is a meromorphic function with maximal deficiency sum. Now

(q \ q q

r, to; J] An f ) = E N(r, to; An, f) = E (N(r, t; ez) + N(r, Te-ni; ez)) i= 1 ' i= 1 i= 1

and also we have ¿(0, ez) = 1 = ¿(to, ez). As r —> to, N(r, t; ez) = N(r, Te-ni; ez) ~ T(r, ez) and with help of Valiron-Mo'honko identity (1.1), T(r, n?=1 An,f) = E?=1 2T(r, ez). Therefore,

rYtTA A 2(1 -

On removing the condition of maximal deficiency sum of f we obtain the follows result:

Theorem 2.3. Let f (z) be a transcendental meromorphic function of order less than one and ¿(to, f) = 1. Then

/ q

^¿(a,f) < ¿( 0j] An, f

aeC ^ i=1

Theorem 2.4. Let f(z) be a transcendental meromorphic function with maximal deficiency sum of order less than one. Then

' 1

r^ro T (r, / )

2(q + l)(q + 2)-l

(1 - ¿(to,/))+ ¿(to,/);

N(r,0-,L(Avf)) ^ 2 -¿(00,/)

llTTl ^ ' 1 _

T(r,L(Avf)) ^ ¿(00; /) +[1(9+ !)(<? + 2) -1](1- ¿(00; /)):

— JV(r, 00; ¿(An/)) ^ [^C? + l)(g + 2) - 1] (1 - ¿(00, /)) mjlim™ T(r, L(Avf)) ^ (2 -¿(00,/))

Corollary 2.3. The deficiency of 0 with respect to L(Anf) is ¿(0, L(A„/)) = 1 — lim <

X(r,L(A„/)) - ¿(co;/) + [!(, + !)(, +2)-l](l-«(oo;/))'

Corollary 2.4. If in the above result we take ¿(to, f) = 1, then we get

jV(r,oo;L(A,/)) ™ T(r, L(Avf))

Moreover, the deficiency of to with respect to L(Anf) is

T (r,L(Anf ))

Remark 2.3. If equality occurs in (i) of Theorem 2.4 whenever q = 1. Remark 2.4. If q = 1, then we can find ¿(0, L(Anf)) = 1 and ¿(to, L(Anf)) = 0. Remark 2.5. If we put q = 1, then Theorem 2.4 will coincide with a combined result of Theorem C and Theorem D.

Example 2.2. Take /(z) = -r^, where r is a complex constant. Then,

ez — t '

q

L(An f ) = £ atA; f

p(ez)

Oi "" *

q '

i=i J] (ez+in - t)

i=0

where p(ez) is polynomial of ez. Hence, L(Anf) ^ 0. Now, ¿(0, f) = 1, ¿(-t, f) = 1,

¿(to, f) = 0. Thus f (z) is a meromorphic function with maximal deficiency sum. Now

q

N(r, to; L(Anf)) = £ N(r, Te-in; ez)

i=0

and also we have ¿(0,ez) = 1 = ¿(to,ez). As r —► to, N(r, Te-in;ez) ~ T(r, ez) and with

s f )) = £ q=o:

help of Valiron-Mo'honko identity (1.1), T(r,L(Anf)) = £q=0 T(r,ez). Therefore,

— jV(r,oo;L(A,/)) ™ T(r, L(Avf)) ^ ■

Again if we remove the condition of maximal deficiency sum of f, then we arrive at the follows result:

Theorem 2.5. Let f (z) be a transcendental meromorphic function of order less than one and ¿(to, f) = 1. Then

^¿(a,f) < ¿(0,L(Anf)).

ae€

3. Lemmas

In this section,we state some lemmas which will be needed in the sequel. Lemma 3.1 [9]. Let f (z) be a meromorphic function with order a (finite) and n € C \{0}. Then,

f(z + rj)\ _ l+£.

m r,

f (z)

where e € R+ \ {0}.

= 0(r°

Lemma 3.2 [8]. Let f (z) be a meromorphic function with order a (< 1) and n € C \ {0}. Then,

fiz + v)^ =0(T(r,f))= S(r,f).

m r.

/ (z)

Lemma 3.3 [8]. Let f (z) be a meromorphic function with order a (< 1) and n € C \ {0}. Then,

N(r, to; f (z + n)) = N(r, to; f) + S(r, f).

Lemma 3.4. Let f(z) be a meromorphic function with order a (< 1) and n € C \ {0}. Then,

1 ' +!)(<? + 2) -1

N(r, to; L(an/)) <

2

N(r, to; /)+ S(r, /).

< We have

L(An /) = E fliA; / = oi Aj / + A2A2 / + ■ ■ ■ + Oq A« /

i=1

Using Lemma 3.3, we deduce that,

N(r, to; Anf) = N(r, to; f (z + n)) + N(r, to; f) < 2N(r, to; f) + S(r, f),

N(r, to; Anf) = N(r, to; f (z + 2n)) + N(r, to; f (z + n)) + N(r, to; f) < 3N(r, to; f) + S(r, f).

Proceeding similarly we find,

N(r, to; Anf) < (q + 1)N(r, to; f) + S(r, f),

and thus,

N(r, to; L(Anf)) < 2N(r, to; f) + 3N(r, to; f) + ■ ■ ■ + (q + 1)N(r, to; f) + S(r, f)

<

(q + 1)(q + 2) - 1

N(r, to; /) + S(r, /).

Hence the lemma. >

4. Proofs of Theorems

Proof of Theorem 2.1. With help of Lemma 3.2 and Lemma 3.3, we deduce from Nevanlinna's first fundamental theorem,

T hn An /

i=1

mr

i=1

flA*/) + N(r, to^A^i/

i=1

/q II Ani/

= m r, ■

i=1

+ N(r, TO^Ani /

i=1

/q q

/ n Ani A q < m(r, fq)+m[ r, 1=1 fq ) + EiV(r, oo; AVif) + O(l)

i=1

1

2

^ m(r, fq) + m

n A^i f

r, ■

i=1

fq

+ £ N(r, to; f (z + n)) + E N(r, to; f) + O(1)

/

i=1

i=1

^ qm(r, f) + m

1 n Ani fX

i=1

fq

+ qN(r, to; f) + qN(r, to; f) + S(r, f)

V

/

< qT(r, f)+ qN(r, to; f) + S(r, f).

Hence,

_T(r'.n>/) -N(roo-f)

lim ry>/} < q + q hm ^^ ^ + _ 5(oo> /)) = ,(2 _ 5(oo; /)). (4.1)

Let {aj : j = 1,2, ...,q} be the sequence of finite deficient values of f (z). We construct a function ^(z) on open complex plane as

q ?

^(z) = nE

1

f - a

¿=1 j=i

where q € Z+. Now, T(r, f — aj) = T(r, f) + O(1) and Ani (f — aj) = Ani f, which implies that nq=i Ani (f — aj) = Hq=1 Anif. Since order of ^(z) is less than 1, from Lemma 3.2, we have

q \ q q

f(z) ^EE

mr

= S(r,f).

¿=i ¿=i j=i Now making use of the above relation we show that,

Tn{r, ip(z)) = m (r, ip(z) f AViA — --) ^ m (r, — --)+S(r,f). (4.2)

V W I! A^i A \ I! A^i f/

i=1

i=1

Since ^(z) is polynomial of degree q2, from Valiron-Mo'honko identity (1.1), Nevanlinna's first fundamental theorem and inequality (4.2), we have,

q2T(r, f) + N^r, 0; n Ani A = T(r, + N (r, 0; n A^ A + O(1) ^ ¿=i ' ^ ¿=i '

= m(r,#z)) + N(r, to; #z)) + N^r, 0; ^ f) + O(1)

^ ¿=i '

/ q \ q q

< m(r, ^(z)) + Wr, 0^ Ani f ) + EE N (r, aj; f) + O(1)

V ¿=i / i=! j=!

^ m r,

i=1j=1

/ q \ q q E[Ani f/ V i=1 7 i=1 j=1

1

i=1

q

i=1

qq

< T( r, [] Anif ) + EEN(r,aj; f) + S(r, f). i=1j=1

Hence,

/9 \ 9 9

q2T(r, f) < Wr^ Ani f + E E N(r, aj; f) + S(r, f), ^ i=1 7 i=1 j = 1 / 9 \ 99 , q x 99

rfr^ Ani f) £ £ N(r,aj; f) T(V, Ü A^ A £ £ N(r,aj; f)

2 ^ V i=l 7 , »=lj = l . v i=l 7 -p— z=lj=l

q * T(r,f) +-T(r,/) + -T{r\fj-

T(r, n Ani A 9 9

< m ;r!n + EE(1-^/))-

T (r,f ) i=1 j=1

Thus,

T(V, ft Ani j , g g T f) i=1 j=1

Therefore we obtain,

T(r, ft A^, , g

i=1

_ > X X I

i=1 j=1

M —^rn— ^ E E /) = E(2 - /)) = - /))• (4-3)

™ i=ij=i i=i

Now combining (4.1) and (4.3), we get

T(r, ft Anif)

¥ —— =q(-2 - /))•

r^ro T (r, f)

At the same time, from Nevanlinna's first fundamental theorem and inequality (4.2), we deduce

q q / q \ / q \

EEm(r,®J)+ Wr,0^Anif j < m(r,^(z)) + Nir,0;^^fl

i=1 j=i ^ i=1 ' ^ i=1 7

j=

9 \ / 9

n Ani f -

i=1

< m f r, -5-i-) + N (r, 0; AVi /) + S(r, /) < T (r, f[ Am /) + S(r, /).

(4.5)

Hence,

99 ££m(r,ai) ¿=1J'=1 <i + .

T(r,ft Ani A T(V, ft Ani

i=1 7 v i=1

It follows that

9 9 , , N(V, O^Ani f)

EEM m(rj) + H5I—^—i

^ ™ T(r, n \j) ™ T(r, n \j) v i=1 7 v i=1 7

9

Using (4.4) we deduce

1 ^ lim

r—y^o

N(V, Ü^Ani f) q q ( ) T( f)

+ EE ÜSI

r(r, n A,/) tijtt™ T(r,f) r^ooTf ß aA

V i=1 / V i=1 /

/ q \ q q

N(V, 0;n Ani ,f)

-— v i=1 7 , i=1j=1

^ lim

r—^

T(V, ft Anif) v i=1 7

q(2 - ¿(to, f))'

Therefore we obtain,

1 ^ lim --+ 1.

r—^

T(r^ Ani f) v i=1 7

Hence,

lim

r—y^o

N r, 0^ Anif

i=1

T(V, ft Ani f) v i=1 7

0.

Therefore,

lim

r—^

N r,0^ Anif

v ¿=i_

r(r, ft A„/) v i=1 7

0.

Proof of Theorem 2.2. Making use of Lemma 3.3 we deduce that q\qq

N r, to; [] Anif = E N(r, to; Anif) ^ 2N(r, to; f) + S(r, f) < 2qN(r, to; f) + S(r, f).

i=i i=i i=i In view of the above inequality, we show

q \ / q

I

=i

t(t, n A,„/) T<r'«

V i=i 7

Now, by Theorem 2.1(1), we have

T (r,f) T (r,f )•

5(2 - ¿(oo, /)) lim --— < 2<K1 - S(oo, /)).

T(r^ Ani f) v i=1 7

q

q

q

Hence,

lim

r^-ro

N(V, to^An- f) V / i VtJ ) < 2(1 -S(

- ))

T(r,n Anif) (2 - )) v i=1 7

Finally, making use of Theorem 2.1(2), we get

Wn Ani f

i=1

<

2(1 — ¿(oo, /)) (2 - 5(oo, /)) *

Proof of Theorem 2.3. To prove the result, consider two cases. Case I. Under the assumption EaeC ¿(a, f) = 0 the theorem is obviously true. Case II. We assume Eaec ¿(a, f) > 0. Let {aj : j = 1,2,..., q} is sequence of deficient values of f (z) where aj € C and q is any arbitrary positive integer. Now from inequality (4.5), we have

99

EEm(r,«j)+ N(r, 0;[]Ani f ) < T( r , [] Ani f ) + S(r,f). i=1 j=1

Then for r —y to, we have

i=1

i=1

99

9 9 9

^ . ,EEm(r,aj) , E Em(r,a,)+ N r,0; n A^f lir-f) " ' ■ „(1) ____tl_1.

T(r, Ü Ani f) v i=1 7

T (r,f)

Now, taking into account the inequality (4.1), we deduce that T (r,f)

T(V, ft Ani f) v i=1 7

lim

r^-ro

lim

r—>oo

T(r^ A„f)V T^>

V i=1 7

T (r,f)

9 9 / 9

EEm(r,aJ) . Mr,0; n Anif

i=1 j = 1 C1\1 I i=1

- -o(l) +

T(r, Ü Ani f) v i=1 7

^ 1

99

E I

¿=1 j=1

E E m(r,aj)

T(r^ An./)V T('./)

v ¿=1 7

- o(1)

+ lim -- ^ 1

r—^^o / 9

T(r, Ü Ani f)

v ¿=1 7

T(r, f) lim --- lim

99

E I

¿=1 j=1

EEm(r,aJ) Mr, 0;IIAni f

T(r^ Anif) v ¿=1 7

T(r, f)

9

n

¿=1

T(r, n Anif)

v ¿=1 7

^ 1

99

E E^(«j ,f) i=li=l_

<7(2 - ¿(to,/))

- M 0,n Ani f) ) < 1-

¿=1

According to our hypotheses that ¿(to, f) = 1 and q is arbitrary, we have

£ ¿(a, f) < ¿(0, H AniA

aeC ^ i=1 '

9

9

Proof of Theorem 2.4. With the help of Lemma 3.2, Lemma 3.3, and Lemma 3.4, we deduce from Nevanlinna's first fundamental theorem,

T(r, L(Avf)) = m(r, L(Avf)) + N(r, oo; L(Avf)) = m (r, + ^ L(A?7/))

< m(r, f) + m (V, ^11) + N(r, oo; L(Avf)) + 0( 1)

< m{r, f)+m (r, + N(r, oo; L(Avf)) + 0{ 1)

f

^ m(r, f) + m[r, | +

(q + 1)(q + 2) - 1

N(r, to; f) + S(r, f)

^ m(r, f) +

1

+ 1)(q + 2) - 1

Hence,

r—ro T(r, f) r—ro T(r, f)

N(r, to; f)+ S(r, f).

iV(r,oo;/)

-(i + l)(i + 2)-l

lim

r—ro T(r, f)

= ¿(to; f) +

(q + 1)(q + 2) - 1

(4.6)

(1 - ¿(to; f)).

Let {aj : j = 1,2,... ,q} be the sequence of finite deficient values of f(z). We construct a function ^(z) on open complex plane as

^(z) = E

1

f — a

where q € Z+. Now, T(r, f — aj) = T(r, f) + O(1) and A^,(f — aj) = A^,f, which implies that An(f — aj) = L(Anf). Since order of ^(z) is less than 1, it follows from Lemma 3.2 that

m{r^(z)L(Avf(z))) =S(r,f).

j=i f — aj

Now with the help of above relation we show that,

1

m(r,ip(z)) = m[ r,ip(z)(L(Ar]f))-

^ m r,

1

+ S (r,f). (4.7)

VJ " (L(Avf)) J ■■■y l.( A,J),

Since ^(z) is polynomial of degree q, from Valiron-Mo'honko identity (1.1), Nevanlinna's first fundamental theorem and inequality (4.7), we have

qT(r, f) + N(r, 0; L(Anf)) = T(r, + N(r, 0; L(Anf)) + O(1)

= m(r, ^(z)) + N(r, to; ^(z)) + N(r, 0; L(Anf)) + O(1)

q

< m(r, ^(z)) + N(r, 0; L(Anf)) + £ N(r, aj; f) + O(1)

j=1

^ m r

1

L(An f)

+ N(r, 0; L(Anf)) + £ N(r, aj; f) + S(r, f)

j=1

q

< T(r, L(Anf)) + E N(r, aj; f) + S(r, f). j=1

1

2

2

Hence, qT(r, f) < T(r, L(Anf)) + j N(r, aj; f) + S(r, f),

99

EN (r,a?-; f) EN (r, a,-; f)

Thus,

Therefore we obtain,

lim nrTL^f] > E Ä(aif /) = (2 - ¿(00, /)) = (2 - ¿(00, /)).

r—ro T (',f) j=1

(4.8)

Now combining (4.6) and (4.8), we have

(2 - ¿(00, /)) < lim T{r^,f)) < ¿(00; /) + r—ro T (r, f)

-(q + l)(q + 2)-l

(1 - ¿(to; f)). (4.9)

At the same time, from Nevanlinna's first fundamental theorem and inequality (4.7), we have

E m(r, aj) + N(r, 0; L(Anf)) < m(r, ^(z)) + N(r, 0; L(Anf))

j=1

< m (r' ¿¿ö) + N{r'0; L(A"/)} + ^ T(r' + /)•

(4.10)

Hence,

E m(r, a7-)

61 , N(r,0;L(Avf))

+

T (r,L(An f)) T (r,L(An f))

< 1 +

S (r,f)

T (r,L(An f))'

It follows that

q

E „.■"(;;?)„, + Ei ^^ < 1 + lim '> '>

j=1

r—ro

T(r,L(Anf)) r—ro T(r,L(Anf))

r—ro T(r,f) T(r,L(Anf))

= 1.

Taking into account (4.9) we estimate

1 > lim

N(r, 0; L(Anf)) , ^ ^ m(r,aj) T(r, f)

+ lim

lim

™ T(r,L(Avf)) ' T(r,f) —ooT{r,L{A^/))

> EE N^°lL[AZf)) +

E ¿(aj ,f) j=1

™ T(r, L(Avf)) ¿(00; /) + + l)(q + 2) - 1] (1 - ¿(00; /)) •

9

Therefore we obtain,

1>mN(rO [L(\f)) +

2 - ¿(to, f)

™ T(r, L(A„/)) ¿(00; /) + [\{q + l){q + 2) - l] (1 - ¿(00; /))'

Hence,

ATM; ¿(A,/)) ™ T(r, L(Avf)) ^

2 - ¿(to, f)

¿(00; /) + [±(<7 + !)(<? + 2) - 1] (1 - ¿(00; /))'

Therefore,

N(r,0;L(Avf)) _ ™ T(r, L(Avf)) ^

2 - ¿(to, f)

¿(00; /) + [±(9 + !)(<? + 2) - 1] (1 - ¿(00; /))"

Again, making use of Lemma 3.3 and Lemma 3.4, we deduce

N(r, to; L(anf)) <

±(q + l)(q + 2)-l

N(r, to; f)+ S(r,f).

Using the above inequality, we show N(r, to; L(anf)) T(r,L(anf))

<

T (r,L(An f)) T (r,f) Now using the result (i) of Theorem 2.4, we have

N(r, 00; /) S(r,/) T(r,/) T(r, /)'

r—ro T (r,L(An f ))

i(i + l)(i + 2)-l

(1 - ¿(to, f)).

Hence,

— A/"(r, 00; L(Avf)) ^ (<? + !)(<? + 2) - 1] (1 - ¿(00,/))

lim <; L2

™ T(r, L(Avf))

(2 - ¿(to, f))

Proof of Theorem 2.5. To prove the result, consider two cases. Case I. We assume EaeC ¿(a, f) = 0, then the theorem is trivially true. Case II. We assume E aeC ¿(a, f) > 0. Let {aj : j = 1,2, ...,q} be sequence of the deficient values of f (z) where aj € C and q arbitrary positive integer. Then from (4.10) we have

q

E m(r, aj) + N(r, 0; L(Anf)) < T(r, L(Anf)) + S(r, f). j=i

For sufficiently large r, we can estimate

T (r,f)

qq

E m(r, aj) v E m(r, aj) + N(r, 0; L(Anf)) j=1 j=1 -o(l) +-

T(r,L(Anf)) ^ T(r,f) T(r,L(Anf))

Now with the help of inequality (4.6), we deduce that

^ 1.

lim

r—ro

lim

r—>00

T (r,f)

T (r,L(An f ))l T (r,f)

E m(r, a7-) N

Ä j;_o(1)\+Ar(r,0;L(A,/))

T (r,L(An f))

^ 1

T (r,f)

' E m(r, aj) j=1

T (r,L(An f ))l T (r,f)

- o(1)

+ ™ T(r, L(Avf))

q

£ m(r, a)

Ihn lim'" +ïï5JV(r'°^(A"/»<l

™ X(r,L(A„/)) ™ T(r,/) ™ Tfr.HVI)

q

£ )

+ (l-5(0,L(A„/))) <1.

¿(00; /) + [±(<7 + l)(g + 2) - 1] (1 - ¿(00; /)) Now, according to our hypotheses that ¿(to, f) = 1 and q € Z+, we have

£ ¿(a,f) < ¿(0,L(Anf)).

ae€

Conflicts of Interest. The author declare that he has no conflict of interest about the publication of the article.

Acknowledgement. I am grateful to Dr. Samten Tamang (Asst. Prof. of The University of Burdwan) and Dr. Nintu Mandal (Asst. Prof. of The Chandannagore Govt. College) for their suggestions to improve the paper.

References

1. Hayman, W. K. Meromorphic Functions, Oxford, Oxford Mathematical Monographs Clarendon Press, 1964, 195 p.

2. Yang, L. Value Distribution Ttheory, Translated and revised from the 1982 chinese original, Berlin, Springer, 1993, 269 p.

3. Zheng, J. H. Value Distribution of Meromorphic Functions, Beijing, Tsinghua University Press, 2010, 308 p.

4. Halburd, R. G. and Korhonen, R. J. Nevanlinna Theory for the Difference Operator, Annales Academiae Scientiarum Fennicae Mathematica, 2006, vol. 31, pp. 463-478.

5. Edrei, A. Sums of Deficiencies of Meromorphic Functions II, Journal d'Analyse Mathématique, 1967, vol. 19, pp. 53-74. DOI: 10.1007/BF02788709.

6. Weitsman, A. Meromorphic Functions with Maximal Deficiency Sum and a Conjecture of F. Nevanlinna, Acta Mathematica, 1969, vol. 123, pp. 115-139. DOI: 10.1007/BF02392387.

7. Fang, M. L. A Note on a Result of Singh and Kulkarni, International Journal of Mathematics and Mathematical Sciences, 2000, vol. 23, no. 4, pp. 285-288. DOI: 10.1155/S016117120000082X.

8. Wu, Z. Value Distribution for Difference Operator of Meromorphic Functions with Maximal Deficiency Sum, Journal of Inequalities and Applications, 2013, vol. 530. DOI: 10.1186/1029-242X-2013-530.

9. Chiang, Y. M. and Feng, S. J. On the Nevanlinna Characteristic f (z + rf) and Difference Equations in Complex Plane, The Ramanujan Journal, 2008, vol. 16, pp. 105-129. DOI: 10.1007/s11139-007-9101-1.

Received April 10, 2021

Abhijit Shaw Balagarh High School,

Balagarh, Hooghly, West Bengal 712501, India, Asistant Teacher E-mail: ashaw2912@gmail.com https://orcid.org/0000-0002-3000-0824

Владикавказский математический журнал 2022, Том 24, Выпуск 1, С. 121-135

О МЕРОМОРФНЫХ ФУНКЦИЯХ С МАКСИМАЛЬНОЙ СУММОЙ ДЕФЕКТОВ И СООТВЕТСТВУЮЩИЕ РАЗНОСТНЫЕ ОПЕРАТОРЫ

Шоу А.1

1 Высшая школа Балагарха, Индия, 712501, Западная Бенгалия, Хугли, Балагарх E-mail: ashaw2912@gmail.com

Аннотация. В этой статье мы имеем дело в основном с характеристической функцией и дефектом мероморфной функции. В качестве эффективного инструмента исследования пользуемся понятием максимальной дефектной суммы (= максимальной суммы величин дефектов) мероморфной функции. Основное внимание уделяется связи между характеристической функцией произведения разностных операторов и характеристической функцией мероморфной функции с максимальной дефектной суммой. В этом же контексте рассматривается разностный полином от разностного оператора. Статья содержит также детальный анализ и обсуждение асимптотического поведения произ-

T (r,n|_i Дщ f) Г N(r,0;n|_! Дщ f)

ведения разностных операторов, таких, например, как пт^-юо-„'г1,, г—, йтг^ш , „q .—-Ьг-,

T (r,f ) T (r,i li_i Ani f )

iimr^oo- i,q—г——-— и др.; аналогичным образом рассмотрен также разностный по-

T (r'i li_ 1 Ani f )

лином разностных операторов. Представлены также некоторые неравенства для нулей и полюсов для Ш=1 Дщ f и Ь(ДП f). По ходу изложения представлены несколько замечаний и следствий, а также даны два примера для надлежащего обоснования наших результатов. Эти результаты усиливают или обобщают результаты З. Ву.

Ключевые слова: трансцендентная мероморфная функция, сумма дефектов, разностный оператор, произведение разностных операторов.

AMS Subject Classification: 30D35, 39A05.

Образец цитирования: Shaw, A. On Meromorphic Function with Maximal Deficiency Sum and it's Difference Operators // Владикавк. мат. журн.—2022.—'Т. 24, № 1.—C. 121-135 (in English). DOI: 10.46698/g4967-8526-0651-y.