Morera’s theorem and functional series in the class of A-analytic functions Текст научной статьи по специальности «Математика»

УДК 517.548.2

Morera's Theorem and Functional Series in the Class of A-analytic Functions

Nasridin M. Jabborov*

National University of Uzbekistan Vuzgorodok, Tashkent, 100174 Uzbekistan

Received 04.05.2017, received in revised form 18.10.2017, accepted 20.11.2017 The aim of this paper is to investigate A-analytic functions in a special case when the function A is an anti-analytic function in a domain. We prove that a continuous function satisfying the integral condition of the Cauchy theorem is A-analytic (an analog of Morera's theorem, Sec. 2). In Sec. 3 we prove an analog of the Weierstrass theorem for functional series of A-analytic functions and the expansion of A-analytic functions into functional series (Sec. 4).

Keywords: A-analytic functions, analog of Morera's theorem, analog of the Weierstrass theorem, expansion of A-analytic functions. DOI: 10.17516/1997-1397-2018-11-1-50-59.

1. Introduction and preliminaries

The paper is devoted to the theory of real-analytic solutions of the Beltrami equation

f (z) = A (z) fz (z) (1)

which is directly related to theory of quasi-conformal mappings. The function A (z) is, in general, assumed to be measurable with \A (z)\ < C < 1 almost everywhere in the domain D C C. Solutions of equation (1) are often referred to as A-analytic functions in the literature.

The solutions of equation (1), as well as quasi-conformal homeomorphisms in the complex plane C, have been studied in sufficient details. Here we confine ourselves to giving the references ( [1,4,5,8-10]) and formulating the following three theorems:

Theorem 1.1 ( [1]). For any measurable on the complex plane function A(z): HA^ < 1 there exists a unique homeomorphic solution x(z) of equation (1) which fixes the points 0, 1, x.

Note that if the function \A(z)\ ^ C < 1 is defined only in the domain D C C, then it can be extended to the whole C by setting A = 0 outside D, so Theorem 1.1 holds for any domain D C C.

Theorem 1.2 ( [4,5]). All generalized solutions of equation (1) have the form f (z) = $[x(z)], where x(z) is a homeomorphic solution in Theorem 1.1, and $(£) is a holomorphic function in the domain x(D). Moreover, if a generalized solution f (z) has isolated singular points, then the holomorphic function $ = f o x-1 also has isolated singularities of the same types.

Theorem 1.2 implies that an A-analytic function f carries out an internal (open) mapping, i.e. it maps an open set to an open set. It follows that the maximum principle holds for such functions: for any bounded domain G c D the maximum of the modulus is reached only on the boundary, i.e. \f(z)\ ^ max \f(z)\, z G G. If the function is not zero, then the minimum

zEdG

principle also holds, i.e. \f(z)\ > min \f(z)\, z G G.

zESG

* [email protected] © Siberian Federal University. All rights reserved

Theorem 1.3 ([8]). If a function A(z) belongs to the class Cm(D), then every solution f of equation (1) also belongs, at least, to the same class Cm(D).

The aim of this paper is to investigate A-analytic functions in a special case when the function A is an anti-analytic function in a domain. We prove that a continuous function satisfying the integral condition of the Cauchy theorem is A-analytic (an analog of Morera's theorem, Sec. 2). In Sec. 3 we prove an analog of the Weierstrass theorem for functional series of A-analytic functions and the expansion of A-analytic functions into functional series (Sec. 4).

The study of A-analytic functions was inspired by their applications in tomography problems. In a series of papers by A. Bukhgeim and S. G. Kazantsev (see [6,7]) the Radon problem is interpreted as a boundary value problem for an infinite-dimensional analog of the equation f — Afz = 0, where f is a function of complex argument z with values in some Banach space X, and A is a linear continuous operator A: X ^ X, ||^^L|| < 1.

A-analytic functions can be applied in the theory of elliptic equations (see [11,16]), when A is a continuous linear operator in a finite or infinite-dimensional space. In papers [11,16] A is a linear continuous operator in X. In case when X = C the function A is a constant.

dA

Let A be anti-analytic, i.e. — =0 in D c C , and such that \A(z)\ < C < 1, ^z e D. We

put

d - d - d d

Da = Si — A(z) 3? a = ^ — A(z) ^

Then according to (1) the class OA(D) of A-analytic functions in D is characterized by the fact that DAf = 0. Since an anti-analytic function is smooth, Theorem 1.3 implies that OA(D) c C ™(D).

Theorem 1.4 (an analog of Cauchy's theorem, see [16]). If f e OA(D)(^\ C (D), where D c C is a domain with rectifiable boundary dD, then

I f (z)(dz + A(z)dz) = 0.

JdD

Now we assume that the domain D c C is convex, and £ e D is a fixed point in it. Consider the function

K(z>0 = 2--. 1 , , (2)

z — £ + f A(t)dT

■7(H,z)

where y(£, z) is a smooth curve which connects points £ and z in D. Since the domain is simply connected and the function A(z) is holomorphic, the integral

I (z) = i A(t )dT Jl(i,z)

does not depend on a path of integration; it coincides with a primitive, i.e. I'(z) = A(z).

Theorem 1.5 ([14]). K(z, £) is an A-analytic function outside of the point z = £, i.e. K e Oa(D \ {£}). Moreover, at z = £ the function K(z, £) has a simple pole.

Remark 1. If a simply connected domain D c C is not convex, then the function

№,£) = z — £ + A(T )dT,

h(H,z)

although well defined in D, may have other isolated zeros except for £: z, £) = 0 for z e P = {£, £1, £2,... }. Consequently, ^ e Oa(D), z, £) = 0 when z e P, and K(z, £) is an A-analytic function only in D \ P, it has poles at the points of P. Due to this fact we consider the class of A-analytic functions only in convex domains.

According to Theorem 1.2, the function ^(z, £) G Oa(D) carries out an internal mapping. In particular, the set

L(£, r)={ z e D: 01

z - £ + A(t)dr

< r

is open in D. For sufficiently small r > 0 it compactly belongs to D and contains the point £. This set is called an A-lemniscate with the center £ and denoted by L(£, r). It is a simply connected domain (see [13]).

Theorem 1.6 (the Cauchy formula, see [13]). Let D C C be a convex domain and G C D be its subdomain with piecewise smooth boundary dG. Then for any function f (z) G OA(G)f) C(G) we have

f (z) = f K(£,z)f (£)(d£ + A(£)dd),z G G. (3)

■JdG

Let A(z) be an anti-analytic function. The following theorem holds, which, as is not difficult to see, without the condition of anti-analyticity A(z) does not hold.

Theorem 1.7. If f (z) G Oa(G) then

df = df G Oa(G).

The proof of the theorem follows easily from the relation that D^df = dDAf, where

d d Da = j- - A(z)— = d - A(z)d. d z dz

In fact, direct calculation shows that DAd = (d — Ad} d = dd — Ad2, dDA = d (3 — Ad) = dz — dA • d — Ad2 =dd — Ad2, since dA = 0 because of anti-analyticity of A(z).

Note that if A(z) is not identically a constant then other derivatives such as d f or DAf are not A-analytic functions.

2. An analog of Morera's theorem

As in the classical case, for A-analytic functions the inverse of the Cauchy theorem holds.

Theorem 2.1. Let f (z) be a continuous function in a simply-connected domain D and the integral of f (z) over any closed smooth curve T that belongs to the domain D be equal to zero, i.e.

j f (z)(dz + A(z)dz)=0. (4)

Then f(z) is an A-analytic function in the domain D.

Proof. Let the function f (z) = u(x, y) + iv(x, y) and A(z) = a(x, y) + ib(x, y). Then condition (4) can be rewritten in the form of contour integrals of the 2nd type:

j) f(z)(dz + A(z)dz) = j) ((a + 1)u — bv)dx + ((a — 1)v + bu)dy+

+ i j) ((a + 1)v + bu)dx + ((a — 1)u — bv)dy = 0.

Hence,

® ((a + 1)u — bv)dx + ((a — 1)v + bu)dy = 0,

j (5)

® ((a + 1)v + bu)dx + ((a — 1)u — bv)dy = 0. We fix a point a e D and consider the following integral

F (z)= i f (z)(dz + A(z)dz), (6)

Jr(a, z)

where r(a, z) is a smooth curve connecting the points a and z e D. According to (4), the integrals (5) do not depend on the path of integration r(a, z). We write the function F(z) in the form

F (z) = U (z) + iV (z), (7)

where

U (z) = ((a + 1)u — bv)dx + ((a — 1)v + bu)dy,

Jr(a, z)

V (z)= ((a + 1)v + bu)dx + ((a — 1)u — bv)dy,

Jria.z)

'r(a, z)

and according to (6), each of these integrals do not depend on the path of integration, and the following equalities hold

dU 1 dU .

-7— = (a + 1)u — bv, -7— = (a — 1)v + bu,

dx dy

dV dV

—— = (a + 1)v + bu, —— = (a — 1)u — bv. dx dy

Hence,

dF _1(dF ,dF\ = 1(dU dV\ i (dU dV\ = . = f

(z = 2\ dX — «dy) = n dxx— dy) + n dyy + dx) = u +iv =t,

dF 1 (dF dF ) 1 (dU dV ) i (dU dV ) . A r ^

JZ = 2\ dX + «dy) = 2 \ dX + dy) + 2{ dy — dx) = au — bv +i(bu + av) = Af.

Since

dF dF

-jT — Att = Af — Af = 0,

d z dz

the function F(z) is an A(z)-analytic function, i.e. F e OA(D). In particular, F e CX(D). dF

According to (8) f = —— and by Theorem 1.7 f e OA(D). The theorem is proved. □

dz

3. Functional series

Lemma 1. Let D c C be a bounded domain with a smooth boundary and f, g e C1 (D x D), then the function

F(z)= f f (£, z)d£ + g(£,z)dZ

J dD

is differentiable with respect to z and the following equality holds

dF =( df (£, z)d£ + dg(£, z)d£

JdD

Proof. Let f(£, z) = fi(£, x, y) + if2(£, x, y) and g(£, z) = gi(£, x, y) + ig2(£, x, y), where £ = Z + in , z = x + iy. Then

F (x,y) = (f! + if2)(dZ + idn) + (gi + ig2)(d( — idn) = JdD

= f (fi + gi)dZ + (g2 — f2)dn + i! (f2 + g2)dZ + (fi — gi)dn.

■JdD JdD

The rule of differentiation of an integral depending on a parameter implies

£ = / + fi) dc + — f dn + i i f + fx) dc + — fi) dn.

dx JdD \ dx dx J \ dx dx J JdD \ dx dx J \ dx dx J

Moreover dF

U f+1) dc+(12 - f) +'L( +t) dz+(H - t) d,

■ G

Using now

d d d d . / d d \ dx dz dz' dy \dz dd J

we have

dF 1 (dF dF) 1 f (df1 dgl \ (dg2 df2) ,

dz = H dX + 'dy) = 2 JdD\ dz + ~z)dZ H dz - -d^)dn+

+ 2 J9d(. dz + dy^ dZ +( dz dy) dn J9d dzd + dgd

We can similarly prove that

dF = df (£,z)d£ + dg(£,z)d£.

JdD

□

Now we consider an A(z)-analytic function f (z) in a simply-connected domain D. We fix a point a G D and a lemniscate L(a, r) = {£: \$(a, £)\ < r} C D. Then we have

Lemma 2. In L(a, r) the following equality holds

dnf (z) = f f (£)

dzn 2ni J z)]n+1

dL(a, r)

(d£ + A(£)dd), n = 0,1,..., (11)

where we recall $(£, z) = £ — z + ^ A(t) dr. Proof. By the Cauchy integral formula we have

f fz) = _L f f (£)(d£ + A(£)dd) =± f f (£)(d£ + A(£)dd)

" 2ni J „ £ — z + f. ,, At) dr 2ni J $(£,z) '

We use the obvious relation

d$n(£, z) n_i d$(£, z) n_i -—-= (^ z)——— = —n$ ^ z)

and

dn ( 1 \ _ dn-1 ( д 1 \ _ dn-1 ( 1 \ _ _ n!

âZ" Vф((, z)J _ dzn-1 Vд^ф((, z)J _ âzn-1 \ф2((, z) ) ~ _ Фп+1((, z) '

We have

df _ д nf _ dn ± f f (Od + A(OdO _ 1 f Яп{ 1

On f <~>—'

âzn 2ni J ф(£, z) 2ni J \Ф(£, z)

L(a,r) L(a,r)

1 f n! , ...n! Г f(0

-J dn{ф^Ъ)f+

-f(trn+A(0d0_é j d+

2ni J Фп+1((, Z)j^'k s 2ni J ^(£,z)]n+1

L(a, r) dL(a, r)

□

Theorem 3.1 (an analog of the Weierstrass theorem). If a series of A-analytic functions in the domain D

f (z) = J2 fn(z),fn(z) G Oa(D), (11)

n=1

converges uniformly on any compact subset of this domain, then

1) f (z) G Oa(D);

2) the series (11) can be differentiated term by term:

oo oo

âf (z)_J2 dfn(z), df (z)_J2 fn(z), DAf (z)_J2 DAfn(z); (12)

=1

3) the series (12) converge uniformly on any compact subset of D.

Proof. We fix an arbitrary simply connected domain G с D. By the hypothesis of the theorem the series (11) converges uniformly in G, i.e. its sum f (z) is continuous in G. We can integrate the series term by term along any closed curve 7 С G:

f(z)(dz + A(z)dz) = Y, / fn(z)(dz + A(z)dz). n=\J Y

Since fn(z) is an A-analytic function in G, then by the Cauchy theorem (Theorem 1.4) all the integrals on the right-hand side are zero. Therefore, the integral of f (z) along 7 is also zero. Morera's theorem (Theorem 2.1) implies that f (z) is A-analytic, which proves statement 1.

We now prove statement 2. We choose an arbitrary point a G D and construct a lemniscate L(a, r) = {\ф(z, a)| < r} с D. According to Lemma 2 we have

_ 1 f f (z)(dz + A(z)dz)

âf _ 2ni I ' -• (13)

iff -—\

dL(a,r) {z - a + I7(z,a) A(t)d'T)

Since the series

f(z) _ £-fn(z)_ 2 (14)

(z - a + f7(z, a) A(T)dT) n=0 (z - a + U, a) A(T)dT)

converges uniformly on dL(a, r), then we can substitute (14) into the integral (13) and interchange the sum and the integral:

dfU* = E n f , fn(z)(dz + A(— )2 = E dfnlz

'dbhr) (z - « + f7(z,a) A(t)dTУ

i.e. df = dfn. Uniform convergence of the series dfn(z) on any compact subset of the

n=0 n=i

domain D follows from Cauchy's formula and from the uniform convergence of the series (11). Similarly, we can prove

df (z) = E dfn(z), DaI(z) = E DAfn(z).

n=1 n=l

We have

df = A(z) df = A(z)d J2fn(z) = J2 A(z) dfn (z) = J2 dfn(z)

n=1 n=1 n=1

and

CO CO

DaI = df - A(z)df = E dfn(z) - A(z)J^ dfn(z) =

n - n

n=1 n=1

= E dfn(z)-J2 A(z) dfn(z) = E [dfn(z) - A(z) fz)] = E Da fn(z).

n=1 n=1 n=1 n=1

C

Since the series J2 A(z)dfn(z) converges uniformly and absolutely inside D, then all the series

n=1

participating in these relations also converge uniformly and absolutely inside D. □

Here it is pertinent to note that from uniform convergence of the series, its differentiability in general does not follow. For this, the series of differentials must also be uniformly convergent.

4. Expansion of A-analytic functions into power series

First we note that the analog of power series for A-analytic functions are the following series

C

J2cjФj(z,a),a G D, (15)

j=0

where cj are constants. The domain of convergence of the series (15) is the lemniscate L(a, R) = {| ф(z, a)I < R}, where the radius of convergence is given by the Cauchy-Hadamard formula:

1

— = Km A / \ Cj \ .

R jV

We show that the series (15) converges absolutely and uniformly inside the lemniscate \$(z, a

R + r

z — a + fY(a z) A(r)dr < R. Let r < R. For \$(z, a)\ = —^— the series (15) converges, and therefore 3n0 : for n ^ n0 the following inequality holds

V^ < 2

r + R'

Then for such n ^ n0 and for 1ф(z, a)| ^ r we have

n n ( 2r )n

icn^(z, aT1 < icn№(z, а)Г < ( r + r) .

Hence, the series (15) can be reduced to a convergent numerical series and it converges absolutely and uniformly in {1ф(z, a)| < r}. There is inverse

Theorem 4.1 (see [14]). If f (z) G Oa(L(a, r)), where L(a, r) = {£ G D : 1ф(£, a)I < r} С D is a lemniscate, then the function f (z) can be expanded into the series in L(a, r):

f (z)=J2 Ckфк(z, a).

(16)

к=0

Coefficients of the series are determined by the formula

Ck =

1 dkf(z)

k! dzk

f (t)

- f

2ni J [ф&а)]

oL(a, p)

k+1

(d£ + A(£)d£), 0 < p < r, k = 0,1,... .

Theorem 4.2. The coefficient of a series J2 Cjфк(z, a) converging in a lemniscate L(a, r),

k=0

r > 0, are uniquely determined by its sum

f (z) = J2 Cjфk(z, a)

(17)

k=0

by the formulas

1 dkf (z)

Ck =

k! dzk

f (t)

- !

oL(a, p)

k+1

(d£ + A(£)d£), 0 < p < r, k = 0,1,... .

Proof. We use formulas (11)

dnf (z)

z ) n!

f (t)

dzn 2ni J [ф(£, z)]n+1

dL(a, r)

(dt + A(0d0,n = 0,1,...,

(18)

and

дфп(z, a)

z

= пфп (z, a)

Substituting into (17) z = a , we find f (a) = c0. We now take the partial derivative of the series (17) with respect to z:

z

and then substite z = a, thus we find

д f (z = ci + 2е2ф(z, a) + 3c^(z, a)2 + ...

(19)

df (z)

z

c1. The series (19) is a series converging

in the lemniscate L(a, r). We take its partial derivative and substitute z = a again to obtain

C2 =

1 d2 f(z)

2 dz2

. And so, in k-th step we get ck = —

1 dk f (z)

k! zk

=a

z=a

z=a

z=a

The second part of the formula

1 dkf(z)

k! dzk

hj № + A(m)

oL(a, p)

follows from (18). □

For completeness of the presentation of the material, we give the expansion of functions into 'Laurent' series.

Theorem 4.3 (Laurent series expansion, see [14]). Let f (z) be A-analytic in a ring of lemnis-cates: f £ OA(L(a, R)\L(a, r)), r < R. Then f (z) admits a 'Laurent' series expansion in this ring:

f (z) = E Cj^(z, a), (20)

k=-w

where the coefficients of the series are determined by the formulas

Ck = 2- J %k+i d + A(0d0, r<P<R, k = 0, ±1, ±2,....

dL(a, p)

The series (20) converges uniformly inside the ring

L(a, R)\L(a, r) = {z £ D: r < \^(z, a)| < R}.

The Cauchy inequalities (see [14]). For the coefficients of this series there hold the following inequalities

^ mM\f (z)\: zk £ dL(a, p)} r < p < R, k = 0, ±I, ±22,.... (21)

pk

References

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z=a

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Теорема Морера и функциональные ряды в классе A-аналитических функций

Насридин М. Жабборов

Национальный университет Узбекистана им. М. Улугбека ВУЗ-городок, Ташкент, 100174, Узбекистан

Цель данной статьи — исследование A-аналитических функций в частном случае, когда функция A является антианалитической функцией в области. Доказано, что непрерывная функция, удовлетворяющая интегральным условиям теоремы Коши, аналитическая функция (аналог теоремы Морера, §2). В §3 доказывается аналог теоремы Вейерштрасса для функционального ряда по A-аналитическим функциям и разложение A-аналитических функций в функциональные ряды (§4).

Ключевые слова: A-аналитическая функция, аналог теоремы Морера, аналог теоремы Вейерштрасса, разложение A-аналитических функций.