CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 1 (2019). С. 120-131.

УДК 517.53

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI

ASHOK RATHOD

Abstract In this paper, we develop the value distribution theory for meromorphic functions with maximal deficiency sum for algebroid functions on annuli and we study the relationship between the deficiency of algebroid function on annuli and that of their derivatives. Let W(z)

be an u-valued algebroid function on the annulus A ( ,R0) (1 < Ro < +ro) with maximal

deficiency sum and the order of W(z) is finite. Then i. limsup =2 - 5o(<x>,W) - 0o(m,W);

ii. limsup

V—

N0 (r> w?j

To (r,W 'j

0;

iii.

1— £o (^,^ j

2— So (<x>,W)

< Ko(W')

< 2(1—60(<x,W))

< 2—s0(^,m) ,

where

K0(W') = limsup

r—<x>

No(r, W') + No(r, ^) To(r, W')

Keywords : Nevanlinna Theory, maximal deficiency sum, algebroid functions, the annuli, etc.

Subject Classification: 30D35

1. Introduction

The uniqueness theory of algebroid functions is an interesting problem in the value distribution theory. The uniqueness problem of algebroid functions was first considered by Valiron, afterwards several uniqueness theorems of algebroid functions in the complex plane C were proved (see [3],[11]). In 2005, A.Ya. Khrystivanvn and A.A. Kondratyuk have proposed the Nevanlinna theory for meromorphic functions on annuli (see [4], [5]) and after this work others have done lot of work in this area (see [8], [12], [13] [36]). In 2009, Cao and Yi [1] studied the uniqueness of meromorphic functions sharing some values on annuli. In 2015, Yang Tan [6], Yang Tan and Yue Wang [7] proved some interesting results on the multiple values and uniqueness of algebroid functions on annuli. Thus, it is interesting to consider the uniqueness problem of algebroid functions in multiply connected domains. By doubly connected mapping theorem [10], each doubly connected domain is conformally equivalent to the annulus {z : r < |z| < R], 0 < r < R < +ro. We consider only two eases: r = 0 R = and 0 < r < R < +ro. In the latter case the homothetv z ^ reduces the given domain to the

Ashok Rathod, Characteristic function and deficiency of algebroid functions on annuli.

© Ashok Rathod 2019.

The author is supported by the UGC-Rajiv Gandhi National Fellowship (no. F1-17.1/2013-14-SC-KAR-40380) of India.

Поступила 26 октября 2017 г.

120

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI 121

annulus A ^-1, R0^ = jz : -- < |z| < R0^ , where R0 = . Thus, in both cases each annulus

is invariant with respect to the inversion z ^ 1 We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and algebroid functions (see [2] and [9]),

Let Av (z),Av-i(z),... ,A0(z) b e a group of analytic functions which have no common zeros

and are defined on the annulus A ^ W ,R0^. 1 < R0 ^ and

'—(z, W) = Av(z)Wv + Av-i(z~)Wv 1 + ... + A\(z)W + Ao(z) = 0. (1)

Then irreducible equation (1) defines a w-valued algebroid function on the annulus A ^, Roj (1 < R0 ^

Let W(z) be a w-valued algebroid function on the annulus A ^-1, R0^ (1 < R0 ^ +<x>), we use the notations

f2n

1 ^ i * ^ i

m(r,W) =— / m(r,Wj) =— / — I log+ [wj (гегв )[d6,

u u 2^ J o

N\(r, W) = -

U!(t,W )

dt,

w ) = - r Arm dt,

v Ji t

Ni ^r,

nJ r,

1

wf ( r

N^ ( r.

N1fc I r,

N 2 ( r

W - a 1

W - a 1

W - a 1

W - a 1

W - a 1

W-a

1 Г1 ni (t, wh)

v J1

r

t

dt,

П2 (t, W—<J^

1 Г ,

V Ji t

1 Г1 & W-a)

F .1

t

dt,

1

^k)

n.

2

(t, -1- )

V ’ -—-J

t

dt,

1 f1 n[k (t, w-a) dt

V J1

r

t

1 Г n2 w-a)

dt,

1

m0(r, W) = m(r, W) + m -,W - 2m(1, W)

+ N 2 r

No(r, W) = N1(r, W) + N2(r, W),

No r

1

NU r,

W - a 1

W-a

N1 r

1

Nf I г

W - a 1

W-a

1

+ Щ r

W - a 1

W-a

where Wj (z)(j = 1, 2,... ,f) is a single-valued branch of W (z), n1(t, W) is the counting functions of poles of the function W(z) in {z : t < |z| ^ 1} and n2(t,W) is the counting functions of poles of the function W(z) in {z : 1 < |z| ^ t} (both counting multiplicity). The symbol n1 (t, w~a) stands for the counting functions of poles of the function w~- in {z : t < |z| ^ 1}

1

r

V

T

V

t

V

122

ASHOK RATHOD

and n2 (t, is the counting functions of poles of the function in {z : 1 < |z| ^ t}

(both ignoring multiplicity). By (t,a,W) we denote the number of zeros of W — a in

{z : t < |z| ^ 1} and n2, (t, a, W) is the number of zeros of W — a in {z : 1 < |z| ^ t}, where zero of order < fc is counted according to its multiplicity and a zero of order ^ к is counted exactly к times, respectively.

Let W(z) be a u-valued algebroid function defined by (1) on the annulus A^,Roj (1 < Rq ^ +ro), as

a e C,

no\ r,

(n жЬ) ="»(n yyh)) • Nq{ r-w—z) =1 r ^b)) •

Ф(z,a),

In particular, as a = 0, we have Nq (r, ^) = 1 Nq (r, and as a = ro, the identity holds

Nq (r, W) = 1 Nq (r, -1^ .He re nQ (r, ) and nQ (r, are the counting function of zeros

of W(z) — a and гф(z, a) on the annulus A ( —, Rq j (1 < Rq ^ +ro), respectively.

Definition 1. [6] Given an algebroid function W(z) on the annulus A^, RoJ (1 < Rq ^ +ro), the function

TQ(r, W) = mQ(r, W) + N0(r, W), 1 ^ r < Rq

is called Nevanlinna characteristic of W(z).

Definition 2. Given an algebroid function W (z) on the ann ulus A^ , RQj

(1 < Rq ^ +ro), the order ofW(z) is defined by

шг) г log Tq(t,W )

a(W) = limsup--------------•

г^+те log r

Definitions. Given an algebroid function W (z) on the ann ulus A^ -1, RoJ

(1 < Rq ^ +ro), the value

SQ(a, W) = liminf

mQ(r, wry)

г^+те TQ(r,W)

is called the deficiency of the function W(z) for the value a. For a = <x>, we let

xf Тлп i- • tmQ(r,W) , i- No(r,W)

^Q(w, W) = llmlnf = 1 — llmSUP •

r^+^ To(r,W) г^+ж To(r,W)

IfSQ,a E Сте, we call a a deficient value of W(z).

Definition 4. Given an algebroid function W (z) on the ann ulus A^ , RqJ

(1 < Rq ^ +ro), the value

Nq(t, ^y)

©q^ W) = 1 — llmsup

г^+те Tq(v,W )

and

N°(r, w_a) N°(r, w_a)

Tq (r, W)

are called the reduced deficiencies of the function W (z) for the va lue a.

dQ(a, W) = llmlnf

г^+те

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI 123

2. Auxiliary lemmata

Lemma 1 (The first fundamental theorem on annuli [7]). Let W(z) be v-valued algebroid function defined by (1) on the annulus A (-A, R0\ (1 < R0 ^ +<o), a G C. Then

Tn r,

(r- жЬ) = m°{r- ub) + N° (r- W-t) = Tn(r-w) + 0(1).

Lemma 2 (The second fundamental throrem oa annuli [13]). LetW(z) be v-valued algebroid function defined by (1) on the annulus A^A, R0^ (1 < R0 ^ +<o), ak, (k = 1,2,..,p) are p

distinct complex numbers (finite or infinite), then we have

(p - 2v)Tn (r,W) ^ V nJ r,—1---) - Nl(r,W) + Sa(r,W)

\ W - ak)

(2)

where Ni (r, W) is the density index of all multiple values including finite or infinite, every т-multiple value is counted r — 1 times and

( w' \ ( W' \

S0(r,W) = m0 (^—J +Y, m° [r,W- aj + °(1)'

The remainder satisfies the identity

S0 (r, W) = О (log T0 (r, W)) + О (log r), outside a set of a finite linear measure if г ^ Д0 = +o, while

1

S0(r, W) = О (logTo(r, W)) + o(log—^)

V R0 - rj

outside a set E such that

dr

< +<o as r ^ Ro < +o.

Je - r

Lrnma 3. [7] Let W(z) be v-valu,ed algebroid function defined by (1) on the annulus A ( A ,R0\ (1 < R0 ^ +<o). If the following conditions are satisfied

r ■ fTo (r,W) limini —--------< oo,

log V

r • <To (r,W) ^ liminf-------1— < o,

r^R0 log (Ro_r)

then W(z) is an algebraic function.

До = +o, Ro < +o,

3. Main results

In the present paper we study the problem on the maximal deficiency sum for algebroid function on annuli as well as the relationship between the deficiency of algebroid function on annuli and that of their derivatives.

Theorem 1. Let W(z) be an v-valu,ed algebroid function defined by (1) on the annulus a(А,Щ^ (1 < ^ ^ +<o). Then the -set of all numbers a G C obeying O0(u,W) > 0 is

countable and Y ®o(a,W) ^ 2.

«ее

124

ASHOK RATHOD

Доказательство. By the second fundamental theorem for algebroid function on annuli we have

Q

(q — 2u) Ta(r,W) О о r

E " о (

г=1 4 7

0 (r.W).

and this implies

(q - 2k) < £

N oG W) , So ir.W \ No(r. ^) So(r. W)

+

i= 1

^ lim

Г—У CO

Since

and

we have Therefore,

To(r,W) To(r,W) ,=1

0o (a. W) = 1 — lim sup

sup

rp ( иЛ +limsuP Tw wV

To(r.W) To(r.W)

N~o(r, w_ai)

To(r.W)

So(r. W) = 0(To(r. W)).

(q — 2v) ^ ^[1 — 0o(ai. W)] ^ q — ^ 0o(<^. W).

i= 1 i= 1

0o(ai.w) ^ 2u.

i= 1

and this holds for all extended complex numbers as q ^ 3, Hence,

^ 0o(a.W) ^ 2.

«e€

and this is a defect relation for algebroid functions on annuli.

Let E = [a G C : 0(a. W) > 0} and we are going to show that that E is countable set. We denote

En = <j a G C : 0(a. W) > — } . n G N.

}

Then, bv the defect relation ^contains at most 2n elements and hence Uc^=1En is a countable set. Let us show that

U~=1^n = E.

In order to do this, we take a G E and hence 0o(a.W) > 0, Bv the Archimedian property, there exist n G N such that

0o(a. W) > —.

n

Thus, a G En C U~=1 En and this yields

E C U~=1En. (3)

On the other hand, given a G U(^=1En, we have a G En for some n G N. Therefore,

and hence, a G E. Now we infer that

0o (a. W) > — > 0 n

U~=i E„c E.

Jn= 1 ^ n

By (3) and (4) we get E = Uc^=1En and therefore, E is a countable set.

(4)

□

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI 125

Remark 1. By Theorem 1, the total deficiency of each, algebroid function W(z) on annuli satisfies the inequality

(5)

^5o(a,W) + 5o(<x>,W) < 2.

«ее

If (5) holds, then we say that W(z) has a maximal deficiency sum.

Theorem 2. Let W(z) be an v-valued algebroid function defined by (1) on the annulus A (W, Д^ (1 < R0 < +ж). Then

To(r, W')

limsup

г^-ж To (r, W)

< 2n — $о(ж, W) — ^(ж, W).

Доказательство. We have

By Lemma 2, the identity

, ( W'\ (

mo(r, W) = mJ r, W— j < mJ r, — \ + mo(r, W) + 0(1).

( W(k) \

{ТЧГ) = S„(r,W)

m0 r,

mo(r, W') < mo(r, W) + So(r, W).

holds true and hence,

We also have

N0(r, W') = N0(r, W) + No(r, W) + Nx(r, W).

By (6) and (7) we conclude that

T0 (r, W') =m0 (r, W') + No (r, W') < mQ (r, W) + m0 (r, ^0 + N0 (r, W'),

<m0(r, W) + No(r, W) + No(r, W) + Nx(r, W) + S0(r, W)

<T0(r, W) + No(r, W) + Nx(r, W) + S0(r, W).

This yields

Therefore,

But

and hence,

T0(r, W') < 1+ Np(r, W) + Nx(r, W) + Sp(r, W)

T0 (r,W) < + To (r,W) + T0 (r,W) + T0 (r,W)'

v To (r,W' )^л. No(r,W)

lim suP^NNN\ <1 + lim suP rp ( ил г^ж и (r,W) To (r,w)

<1 + 1 — 0o(ж, W) < 2 — ©o(ro, W)

d0 (ж, W) + во (ж, W) < 0o(^, W)

To (r, W')

lim sup

< 2 — ^(ж,Ш) — во(ж,Ш).

(6)

(7)

To (r, W)

The proof is complete, □

Theorem 2 yields the following corollary,

Torollary 1. Let W(z) be an v-valued algebroid function defined by (1) on the annulus A (, До) (1 < Ro < +ж). Then

T0 (r, W')

limsuP

г^ж To (r,w)

< 2u — do (ж, W) — $o (ж, W).

126

ASHOK RATHOD

Theorem 3. Let W(z) be an v-valued algebroid function defined by (1) on the annulus

A (,Ro) (1 < Ro ^ +ro) with a maximal deficiency sum and of a finite order. Then

i. limsup =2 - 5о(ж, W) - 9о(ж, W);

N0 Д )

To(r,W;)

гг. limsup ;) = 0;

■■■ 1~г0(те,Ж) ^ ^ (w/) < 2{lS0{<x,W)) ...uprp

m- 2-Sa(oo,w) ^ N°(w ) ^ 2-(oo,m) , wnere

2So(<x,W) ^ Л0

2So(<x,W)

No(r,W/) + No(r, -ф)

To(r, W/)

K0(W/) = limsup

v—

Доказательство. We have

To(r, W/) =mo(r, W/) + No(r, W/) ^ mo(r, W) + r, ^ + No(r, W/)

^mo(r, W) + No(r, W) + No(r, W) + W(r, W) + ^o(r, W)

^mo(r, W) + No(r, W) + Wo(r, W) + 2(^ - 1)To(r, W) + S'o(r, W) ^(2^ - 1)To(r, W) + No(r,W) + So(r,W) ^ 2uTo(r,W) + So(r,W). By Lemma 2 we get

So(r, W(k)) = О (log rTo(r, W(k))) = О (log rTo(r, W)) = So(r, W)

and hence

W(k)

f W(k) \

m'0 (Г'Й^й) = S0(r-W)

holds for each positive afif We let

F M = E

^ W(z) - a[i] ■

г=1 4 7

Then, as in [11], we have

m(r

b,F) + 0(1) fi E

ml r

i= 1

P

1

™(V) fiE

V ' i= 1

ml r,

W(z) - a[i] 1

(8)

(9)

(10)

W(z) - a!*]

In fact, (10) holds if p = 1 If p fi 2, we let

6 = mini=j |а[г] - а^\.

It is Obvious that 5 > 0. Given a fixed z, there exist k E {1,2,... ,u} and г E {1, 2,... ,q} such that

5 5

К - «И| < - « j,

and the inequality

of

\wk(z) - a[3]\ fi \ащ - a[3]\ - \wk(z) - | fi

holds true for i = f. Therefore, the set of points in d Cr, where Cr = {z : \z \ = r}(r = r or r = 1) obeying \wk(z)-a^ \ < 2- is either empty or each two such sets are mutually disjoint for different

1

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI 127

i. In any case

-1 f ’ log+ |F(re*)|сЮ J-1 £ f log+ |F(re*)\M

2 Jo 2 i=l J \wfc-aW\< 2^ ,\z\ =r

> 2- E/ iog+

2Ж i=l JK-“w\<2,\z\=r

\ Wk (reie) — aW|

dd.

Since

- E f

2 i=i JК-“[i]\<2|,\z\=r

log

+

| Wk (reie) — aM |

-redd

1

m r,

> m r

1

W(z) — J 2n 1

iL

i=l J\^fc-\^2^,\z\=r

* u,_„bg+ |wk(rew) — aN|

dd

W (z) — a И

i + 2<i — Q log+

we obtain that

1 .v . 1 ('2ж 1

m(r,F)=—V"1— log+ |F(гегв^dd ^ -

^ t=i 2nJo u

EE m{r-wwzss)

fc=i j=i v w 7

— log+ 2_

W(z) — aF) v 5

=E m{r,wuhA)

— — log+ ^.

W (z) — а[г] J v 5

Now relation (10) follows the above inequality in the case r = r and r = -. Since

1

m(r, F) =m(r, W (fc)F) + m\r

E m (

W(k)

r / W (fc) \ / -

° m[r’W—Ml + mir'

i=l

) +

and

(H =m((t)?) + ^-.^)

A (1 W(k) \ (11 \

^E SiwJ + )

we get

,A ( W(k) \ ( 1 \

mo (r, F) ^ г, + mo (y )

(11)

It follows from (8), (11) and Lemma 3,1 that

p

£mo f

i=l '

W(;/ — aw) «m°(+ So(r,W)

ZFo(r, W(fc)) — No F^) + S0(r, W).

(12)

1

1

128

ASHOK RATHOD

Thus,

PT°£ W) « £ N„(r, w(z)_ a,„) + To(r, W(‘>) - N„(r, -L) + S0(r, W).

Bv (13) we obtain

pTo^ w) < £ N° (", w(г/_ a[,l) + ЗДп ^^ (", w) + ^ lv)■

and hence,

p < '

S„(r, W)

+ £(1 _ * (^,W)) + lim inf ^ )

г^ж l0(r,W) j' г^ж lo(r,W)

=p + lim sup

T0(r, W') To(r, W)

£ <5o(aW ,W )■

i=1

Therefore, we have

J]£o(a[tl,W) ^ lim

To(r, W')

i= 1

Hence, identity (8) and Lemma 2 imply that

To(r, W')

limsup

ж To (r, W)

sup гг t ил ■ г^ж lo(r,W)

^ 2u _ 8o(<x, W).

As p is arbitrary, we combine (17) and (16) to have

To(r W') To(r W')

2u _ 8o(<x, W) ^ lim inf . ’ т ^ limsup . ’ T ^ 2v _ 8o (ж, W),

that is,

lim

To(r,W) To(r, W')

To(r, W)

^ 2u _ 8o(<x, W).

г^ж To(r, W)

Given e > 0, we choose q sufficiently large so that

p

8o(a[^, W) > 2v _ ^(ж, W) _ e.

i= 1

For these q, inequality (12) implies

t No(r, ^ . To(r,W) • tm° w b)-“M)

lim su^ i u/л + lim inf rr ( un\ lim inf------r ( ил-----

г^ж To(r,W') г^ж To(r,W') г^ж To(r,W)

^ 1 + lim sup

Sp(r,W) To (r, W') ■

Thus, from (18)-(20) we deduce

T ^o(F 177) ^ £

limAUp ШЩ K 2, _ So (ж, W) •

Since e > 0 is arbitrary, we have

r No(r, 177) 0

limsuD = °.

г^ж To (Г, kK )

V—YOO

V—too

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI 129

And since No(r, W') < 2dNo(t, W), we get

N0(r, W') T0(r,W') < 2 N0(r, W)

To(r,W') To(r,W) < UTo(r,W) .

Now it follows from (18) and (23) that

N0(r W')

(2d — So(<x>, W)) limsup _ . ’ < 2d(1 — So(<x>, W)).

T0 (r, W')

(23)

(24)

Relations (22) and (24) imply

Therefore, we obtain

No(r, W') + No(r, Д) < 2^(1 — 6о(ж, W)) imSUP To(r,W') < (2d — 8o(^,W))'

K0(W) < 2"(1 — ))

(2d — 5o(<x, W)) ‘ Furthermore, we have No(r,W) < No(r.W'). By (18) we infer that

W) < (2, — ))No(r-W')

(25)

To(r,W)

Thus,

v No(r,W' )^

lim sup —ttt— A

lim sup

To(r, W')'

No(r,W) 1 — 5o(m.W)

To(r,W') (2d — 5o(<x>,W)) To(r,W) 2d — 5o(<x>,W)'

Hence,

Ko(W') <

Due to (25) and (26) we have

1 — £o(ro.W)

1 — £o(ro.W)

2d — 8o(x>, W)

2^ (1 — 5o(m,W))

(26)

2d — So(x>, W)

< Ko(W') <

(2d — 8o(<x, W)) '

□

Theorem 4. Let W(z) be an D-valued algebroid function defined by (1) on the annulus A ^(1 < Ro < +ж) of finite order and 8o (ж, W) = 1. Then

Y 5o(TW) < 5o(0,W').

«e€

Доказательство. If

Y 6o(a,W ) = 0,

«e€

Theorem 3 is valid in this case. In what follows we assume that ^ So(a, W) > 0. Let [аД be

«e€

a sequence of distinct complex complex numbers in C containing all the finite deficient values of W(z). For each positive integer q the inequality

£ m<°(r- wpy—y;) + Wo (r' w) < To(r-w'>+s°<r-w>

fl—1

V—too

1

130

ASHOK RATHOD

holds for any q finite complex numbers in a«. Therefore, we have

No < r> W) To (r, W)

+

£ mo (г, щТ^) \

To(r,W') To(r,W')

V

To(r,W)

- o(1)

C 1,

7

as r —— то. Hence, by inequality (18) we obtain that

1 ^ lim sup

V——

No (r, hr) To (r, W)

To(r,W') + To (r,W)

(Yi ^0 (г, щ^) \

M=1 4 7

To(r, W)

- o(1)

7 J

^ lim sup

Г—У CO

^0 1г,т1г

_____wff_ +y . f 70 (o ^)

To(r,W') + ^1^ To(r,W')

(9 V

^ mo (o WWVff) 1 4 7

T0(r, W)

\

- 0(1)

7

^ lim sup

^0 [r, -1

"" ' + lim inf ААЩ lim inf

£m« (r'

. r n=1 4 7

r—~ To(r,W') -—1 To(r,W') -—1 To(r,W)

^ (V мЛ 50 (a^,W)

^ У’ W ) + ^=1_________

^ limSUP To (r,W') ' 2u - 50 (to,W)‘

Since g is arbitrary and 0o (то, W) = 1, we have

^ 0o (a,W) C S0 (0,W').

«e€

□

REFERENCES

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Ashok Rathod,

Department of Mathematics,

Karnatak University,

Dharwad-580003, India E-mail: [email protected]