On a class of mappings garmonic in the half-plane Текст научной статьи по специальности «Математика»

Trudy Petrozavodskogo gosudarstvennogo universiteta

Seria “Matematika” Vypusk 4 1997

UDK 517.54

ON A CLASS OF MAPPINGS HARMONIC IN THE HALF-PLANE

Z. J. Jakubowski, A. Lazinska

Well known is the class of typically-real functions holomorphic in the unit disc |z| < 1, introduced by W. Rogosinski in 1932 ([6]).

There were also investigated classes of typically-real functions harmonic in the unit disc ([1,4]). Whereas in the present paper we consider some class of typically-real functions harmonic in the right half-plane. The results are based on the considerations concerning typically-real functions holomorphic in this half-plane ([3,5]).

§ 1. Definition and basic properties of the class T?

Let n+ = {z e C : Re z > 0} and let first H = H(n+) be the class of all functions f holomorphic in n+ and such that

where af is some complex number.

Let TR be a subclass of functions of H which take real values on the positive real half-axis only, that is

Evidently, for f e TR, in (1.1) we have af e R.

For the class TR, we have

Proposition 1. ([3]) A function f e H belongs to the class Tr if and

lim (f (z) - z) = af

(1.1)

f (z) = f (z) ^ z = z, z e n+.

(1.2)

only if

(1.3)

© Z. J. Jakubowski, A. Lazinska, 1997

Let now H = H(n+) denote the class of complex functions F harmonic in n+ and satisfying normalization condition (1.1). As we know, each function F harmonic in n+ is of the form F = h + g, where h, g are some functions holomorphic in n+, because n+ is a simply connected domain.

Let next T? denote the class of functions F e H, F = h + g, such that Im F satisfies condition (1.3) and

lim Re g(z) = ag, ag e R. (1.4)

n+3z^^

The class T? will be called a class of harmonic functions typically-real in n+.

It follows from normalization condition (1.1) and from (1.3) that aF e R for F e T^.

Example 1.1. Let Fi be a function of the form

Fi(z) = z-------+ —, z e n+. (1.5)

Function (1.5) satisfies conditions (1.1), (1.3) and (1.4), thus F1 e TH.

Directly from the definition of the class T- we have

Proposition 2. If F = h + g belongs to the class T—, then the functions Fn(z) = F(z) + n, n e R, Fs(z) = F(z + S), S > 0, Fp(z) = , p > 0,

z e n+, belong to TH.

Let us notice that

Tr C TH.

Indeed, if f e TR, then f = h + g, where h harmonic in n+, with that we also have (1.3), definition of TH, this proves inclusion (1.6).

We next have

Proposition 3. Let f e TR be a function such that Re f (z) > 0, z e n+, and F e T?. Then the function k = F o f belongs to the class T^.

Proof. Let f e TR, Ref(z) > 0, z e n+ and F e TR-. Since F = h + g is harmonic in n+, the function k = F o f = h o f + g o f is harmonic in n+. Moreover, the function k satisfies condition (1.3) because we have

(1.3) for f and F.

(1.6)

= f, g = 0 is a function (1.4). In virtue of the

Let

af = lim (f (z) — z), aF = lim (F (z) — z),

n+3z—n+

ag = lim Re g(z). Hence lim f (z) = to, therefore

n+3z—n+3z—

lim Re g(f (z)) = ag and lim (k(z) — z) = aF + af.

n+3z—n+3z—

Consequently, the function k = F o f satisfies all the conditions of the definition of the class T^.

Before the next example let us denote by R a subclass of functions f e H mapping n+ conformally onto a domain convex in the direction of the positive real half-axis. The class R is called a class of functions convex in the direction of the real axis in n+ (see [7]). We have

Theorem A ([7]). A function f e H belongs to the class R if and only if

Ref'(z) > 0, z e n+. (1.7)

Of course, by (1.7), all functions of the class R are univalent ([2], p.

88).

Let next Rr denote a subclass of R of functions which take real values for z = g e n+. Obviously,

rr C 7R. (1.70

Example 1.2. Let f1 be a function defined by the formula

fi(z; a, c) = z + a-, z e n+, c > 1, ac > 1. (1.8)

It can easily be proved that: a) f1 e Tr, b) f1(n+) C n+, c) f1 e Rr.

From Proposition 1.3 and the above example it follows that if F e T^, then f o fi e TR*.

The following proposition holds.

Proposition 4. Let F = h + g e iR. Then the function f = h — g belongs to the class TR.

Proof. If F = h + g e TR, then, of course, the function f = h — g is holomorphic in n+. Moreover, Im f = Im F, so, by (1.3), the function f

satisfies condition (1.2). Furthermore,

f (z) — z = h(z) — g(z) — z = h(z) + g(z) — z — 2Re g(z)

= F(z) — z — 2Reg(z), z e n+.

Since F satisfies condition (1.1) and g satisfies condition (1.4), from the

above equalities we get lim (f (z) — z) = aF — 2ag =: af e R. Hence

n+

the function f belongs to TR.

We also have

Proposition 5. Given a pair of functions h, g holomorphic in n+ and such that f = h — g e TR, where g satisfies condition (1.4), the function F = h + g belongs to TR.

Proof. Let the functions h, g satisfy the assumptions. Then the function F = h + g is harmonic in n+; besides, we have (1.4). Moreover, Im F = Im f, so, in virtue of Proposition 1.1, the function F satisfies condition

(1.3), and

F(z) — z = f (z) — z + 2Reg(z), z e n+.

According to the fact that we have (1.1) for f and (1.4) for g, the function

F satisfies normalization condition (1.1). Consequently, F e TR.

In particular, we obtain

Proposition 6. If f e TR, then the function F(z) = 2Re z — f (z), z e n+, belongs to the class TR.

Proof. Let f e TR. Let us set f = h — g where h(z) = z, g(z) = z — f (z), z e n+. Of course, the functions h, g are holomorphic in n+ and g satisfies condition (1.4), which follows from the fact that f satisfies (1.1). Therefore, by Proposition 1.5, the function F = h + g belongs to the class TR, with that F(z) = h(z) + g(z) = 2Rez — f(z), z e n+, which ends the proof.

The proposition below is also true.

Proposition 7. If F = h + g e TR, whereas w is an arbitrary function

holomorphic in n+ and satisfying condition (1.4), then

Fw = h + w + g + w (1.9)

is a function of the class TR.

Proof. Let F = h + g e TR and let w be a function holomorphic in n+ and satisfying (1.4). Of course, the function Fw of form (1.9) is harmonic in n+ and Im Fw = Im F, thus we have (1.3) for the function Fw. Furthermore,

Fw(z) — z = F(z) — z + 2Re w(z), z e n+.

Since F satisfies condition (1.1) and w - (1.4), it follows that Fw also satisfies (1.1). Hence Fw e TR.

. Let F = h + g e TR. We shall prove that the function

F1 (z) = F(z) + 2Re(h(z) — z), z e n+,

also belongs to the class TR.

Let us first notice that, for a function F = h + g e TR, the function h1(z) = h(z) — z, z e n+, is holomorphic in n+ and satisfies

condition (1.4). Indeed, lim Re h1(z) = lim Re (h(z) — z) =

n+ n+ — ^

lim [Re (F(z) — z) — Re g(z)] = aF — ag e R. From this, by Proposi-

n+3z—^

tion 1.7, the function

F 1(z) = h(z) + h1(z) + g(z) + h1(z) = F(z) + 2Re (h(z) — z), z e n+,

as well as F, belongs to the class TR.

Similarly, each function

F^(z) = F(z)+2pRe(h(z) — z), z e n+, p e R,

belongs to TR.

Example 1.3. Let F2 be a function of the form

F2(z; a, b, v) = z + a + —, z e n+, a e R, b > 0, v e (0, 2], 1v = 1. zv

(1.10)

We can prove that function (1.10) belongs to the class TR. Moreover, by Proposition 1.7, the function

Fw (z; a, b, v) = z + a + w(z) +—^ + w(z), a e R, b > 0, v e (0, 2], 1v = 1, belongs to TR if the function w satisfies appropriate assumptions.

In the case b > 0, v > 2 the function F2 of form (1.10) does not satisfy condition (1.3) and it is not a function of TR.

We shall prove

Proposition 8. The class TR is convex.

Proof. If Fk = hk + gk e TR, k = 0,1, then the function F\ = AF1 + (1 — A)F0 = Ah1 + (1 — A)ho + Ag1 + (1 — A)go, A e (0,1),is harmonic

in n+. We also have lim (F\(z) — z) = Aa1 + (1 — A)a0, where n+

ak = lim (Fk(z) — z), k = 0,1. Furthermore, lim Re[Ag1(z) +

n+ n+

(1 — A)g0(z)] = Aa1 + (1 — A)a0 where ak = lim Regk(z), k = 0,1,

n+

and Im F\ = AF1 + (1 — A)Im F0. Hence, in virtue of the definition of the class TR, we infer that F\ e TR for any A e (0,1), which gives the assertion.

§ 2. Properties of the class TR- following from its relationship with the class TR

In this part of the paper we use the known properties of the class TR to obtain the respective consequences for functions of the class TR.

Proposition 9. If F = h + g e TR, then

h(n)(z) + g(n)(z) = h(n)(z) + g(n)(z) for z = g > 0, n = 0, 1, 2,.. . (2.1)

Proof. Let F = h + g e TR. By Proposition 1.4, the function f = h — g belongs to TR and, in virtue of Theorem 1 from [3], we have h(n)(z) — g(n)(z) = h(n)(z) — g(n)(z) for z = g > 0, n = 0,1, 2,..., which is equivalent to (2.1).

Moreover, we have

Proposition 10. Let F = h + g e TR. Then

F(g) = F(z), z e n+, (2.2)

if and only if

h(n)(z) — g(n)(z) = h(n)(z) — g(n)(z) for z = g > 0, n = 0, 1, 2,.. . (2.3)

Proof. Let F = h + g e TR. For any z e n+, there exists z0 = g0 > 0 such that z e U0 = {z e C : |z — z0| <z0}. In virtue of the holomorphy

of the functions h, g in n+, we get

F(z) = h(zo)+gta) + £ ^-^z-zo)” + £ z e Uo,

' n! ^ n!

”=1

”=1

______ o h(”)(z )

F(- ) = h(zo) + g(zo) + E------------------ *o)”

n!

+ E

”=1 ”!

-(z - zo)”, z e Uo,

F(z) = h(zo) + g(zo) + £~i (- - zo)”

g(n)(

I o gW(zo)

-Z-/”=1 ”!

(z - zo)”, z e Uo.

(2.4)

(2.5)

If z = zo + reiV, ^ e (-n, n], r e [0, zo), then, by (2.4), (2.5), we obtain

F(z ) = h(zo) + g(zo) + ^ (h(n)(zo) + g(n)(zo)) cos +

”=1

i(g(n)(zo) - h(n)(zo)) sin n^

oo ”

F(z) = h(zo) + g(zo) + ^ (h(n)(zo) + g(n)(zo)) cos n^ +

”=1

i (g(n)(zo) - h(n)(zo)) sin n^

(2.6)

(2.7)

By Proposition 2.1, conditions (2.1) hold.

If (2.2) takes place, then, in virtue of (2.6), (2.7) and of the arbitrariness of z and zo, we obtain (2.3). Whereas if conditions (2.3) take place, then, in view of (2.1) and (2.6), (2.7), we have (2.2), which completes the proof.

. If relations (2.1) and (2.3) are satisfied simultaneously, then

h(n)(z) = h(n)(z), g(n)(z) = g(n)(z) for z = - > 0, n = 0, 1, 2,. ..,

(2.8)

so the functions h, g and also their succesive derivatives take real values on the half-axis z = - > 0. Of course, if we have (2.8), then equalities

(2.1), (2.3) hold. What is more, conditions (2.8) are analogues of the fact

so

that if a function is holomorphic and typically-real in the unit disc, then it expands in a power series with real coefficients.

Proposition 11. Let F = h + g e TRl. Then

Imh;(z) = Img;(z) for z = - > 0, (2.9)

Re h;(z) > Reg;(z) for z = - > 0. (2.10)

Proof. Let F = h + g e TRH. Then, in virtue of Proposition 1.4, the

function f = h - g belongs to the class T. and, according to the respective

property of functions of the class T. ([3, Prop. 4]), we have h;(z) - g;(z) >

0 for z = - > 0. From the above inequality we directly obtain relations

(2.9) and (2.10).

We also have

Proposition 12. If F = h + g e Tr, then |h(n)(z) - g(n)(z

n!(h'(z) - g'(z))

and

< —- for z = - > 0, n = 1, 2,..., (2.11)

.Urn h(:;(zi - g(”1<z) =0, „ = 2,3,... (2.12)

.=z^o h;(z) - g;(z)

Proof. If F = h + - e TR^, then the function f = h - g belongs to TR. Hence, by the respective property of the class TR ([3, Prop. 5]), we get

(2.11). Obviously, (2.12) follows directly from (2.11).

Proposition 13. If F = h + g e TH and

lim (h(z) - g(z)) = 0, (2.13)

.=z^o+

then

Re h(z) - g(z) > 0, z e n+. (2.14)

z

Proof. If F = h+g e satisfies condition (2.13), then, for the function f = h - g e Tr, in view of Theorem 6 from [3], we have Re f(Zz) > 0, z e n+, which is equivalent to (2.14).

§ 3. Remarks on the compactness of the classes TR, TH

It is known ([3, Th. 5]), that the class T. is not compact. Using the

proof of this fact and inclusion (1.6), we obtain

Proposition 14. The class is not compact.

The proof of the non-compactness of the class TR was based on the construction of a sequence {f”}”£N of functions of the class TR, divergent uniformly on compact subsets of n+ to infinity.

There arises a question: does there exist a sequence of functions of

the class TR, convergent uniformly on compact subsets of n+ to a finite function f e T.? An analogous question concerns the class T^. The partial answer is given by

1. Let (f”}”£N be a sequence of functions of the class TR, convergent uniformly on compact subsets of n+ and on some set DR = {z e n+ : |z| > R}, R > 0, to a finite function f. Then f e TR.

Proof. Let us consider a sequence {f”}”£N satisfying the assumptions of Theorem 3.1. In virtue of the theorem of Weierstrass, the limit function f is holomorphic in n+.

It appears that the function f satisfies normalization condition (1.1). Indeed, let an = lim (fn(z) - z), an e R, n = 1, 2,... The point

n+3z^o

to is an accumulation point of both n+ and Dr, so, in view of the uniform convergence of the sequence {f”}”£N on DR, by the known theorem on sequences of functions, the sequence {a”}neN is convergent and lim (f(z) - z) = lim a” =: a. In virtue of the definitions of the sets

n+ and Dr, we have

+lim (f (z) - z) = lim (f (z) - z) = a,

which gives condition (1.1) for the function f, so f e H.

Let us next notice that since the functions fn, n = 1, 2,..., satisfy condition (1.3), therefore

> 0 if Im z > 0,

Im f (z)< =0 if Im z = 0, z e n+, (3.1)

<0 if Im z < 0.

But if there existed a point zo e n+, Im zo > 0, such that Im f (zo) = 0, we would have Im f (z) = 0 for z e n+, Im z > 0, by the minimum principle for harmonic functions. In view of this fact and of the holomorphy of

the function f, there would be Im f (z) =0, z e n+, so f (z) = c, z e n+ where c e R by (3.1). Hence we would have a contradiction to the fact that f satisfies (1.1). Therefore Imf(z) > 0 for z e n+, Imz > 0. In an analogous way one can prove that Imf (z) < 0 for z e n+, Imz < 0. Summing up, we infer that the function f satisfies condition (1.3), so, in view of Proposition 1.1, it belongs to the class TR.

We shall next prove

. Let F” = h” + -” e Tr , n = 1, 2,..., and let the sequences {F”}neN, {Re g”}”£N be uniformly convergent on compact subsets of n+ and on some set Dr = {z e n+ : |z| > R}, R > 0, to the finite functions F, y>,

respectively. Then F e T?.

Proof. Let functions F” = h” + -” e T?, n =1,2,..., satisfy the above assumptions.

Let us first notice that, analogously as in the proof of Theorem 3.1, we

can state that the complex function F satisfies normalization condition

(1.1).

Since F” = h” + -” e T?, n = 1, 2,..., therefore lim Re g”(z) =

n+ — ^

«”, n = 1, 2,... In virtue of the uniform convergence of the sequence {Re g”}”£N in Dr, proceeding as before, we obtain that the sequence {«”}”£N is convergent and

lim <£>(z) = lim <£>(z) = lim a” =: a e R. (3.2)

n+3z—^ Dr9z—w ”—^

Of course,

^(z) e R, z e n+. (3.3)

We shall prove that the function F satisfies condition (1.3). As it is known from Proposition 1.4, f” = h” - g” e TR, n = 1, 2,... Moreover, f” = F” - 2Re g”, n = 1, 2, . . . From the assumptions about the sequences {F”}”£n, {Reg”}”£N it follows that the sequence {f”}”£N satisfies the assumptions of Theorem 3.1, so the limit function f = F - 2^> of this sequence belongs to the class TR. Hence and from (3.3) we have

( > 0 if Im z > 0,

Im F(z) = Im f (z) < = 0 if Im z = 0, z e n+,

y < 0 if Im z < 0.

Thus F really satisfies condition (1.3).

It remains to prove that F is a function harmonic in n+, i.e. F = h+g where h, g are some functions holomorphic in n+, and that the function g satisfies condition (1.4).

Let us observe that the functions Re g”, n =1, 2,..., as real parts of the functions holomorphic in n+, are real functions harmonic in n+. From this, in view of the uniform convergence of the sequence {Re g”}”£N on compact subsets of n+ to the function f and of the respective properties of harmonic functions, f is a real function harmonic in n+. n+ is a simply connected domain, so there exists a real function 0 harmonic in n+, conjugate to the function f. Thus we have F = f + 2f = Re f + f +

i(Imf + 0) + f - *0, so F = h + g where h = Re f + f + i(Imf + 0), g = f + *0. The functions f, 0 are harmonic 0 and mutually conjugate in n+, therefore g is holomorphic in n+, with that f = Reg and (3.2) holds. Furthermore, since f is a function holomorphic in n+, Re f and Im f are harmonic functions mutually conjugate in n+. In consequence, the functions Re h = Re f + f, Im h = Im f + 0 are a pair similar to f, 0, so the function h is also holomorphic in n+. From these considerations it follows that the function F is a complex function harmonic in n+ and

(1.4) holds.

In virtue of the facts presented above we infer that the function F belongs to T?.

§ 4. The class SR

Let S? denote a class of functions F = h + g e H univalent in n+ and satisfying condition (1.4) and equalities (2.1), (2.3), thus in view of Remark 2.1, conditions (2.8).

. Let us notice that, by the known lemma ([3, Lemma 1]), in order that, for a function F = h + g e H, relations (2.8) hold, it is sufficient that there exists a point zo = go > 0 such that h(”)(zo) = h(”)(zo), g(”)(zo) =

n = 0, 1, 2, . . . , because the functions h, g are holomorphic in n+. This fact can be used in the definition of the class S?.

We shall prove

1. The inclusion

sk c Tk (4.1)

takes place.

Proof. Let F = h + g e S?. In order to state that F e T?, it suffices to show that condition (1.3) holds.

For the function F e S?, we have (2.1) and (2.3), so F(z) = F(z) for z = g > 0 and F(z) = F(g) for an arbitrary z e n+, which follows from the analogous considerations as in the proof of Proposition 2.2. From the univalence of the function F we deduce that F(g) = F(z) if and only if z = g > 0. From the above and by normalization condition (1.1) for F we

have lim (Im F(z) - Im z) = 0. From this, in particular, for any fixed

n+ — ^

xo > 0, lim (Im F(xo + iy) - y) = 0, so there exists zo = xo + iyo,

y——+^,xo >o

xo > 0, such that ImF(zo) > 0. Therefore, in virtue of the continuity of the function F in n+, if Im z > 0, z e n+, then Im F(z) > 0, whereas if Im z < 0, z e n+, then ImF(z) < 0. This means that the function F satisfies condition (1.3) and, in consequence, F belongs to the class T?, which completes the proof.

Example 4.1. Let us come back to the function F2 of form (1.10). For b = 0, we have F2(z; a, 0, v) = z + a, z e n+, so it is a function of the class S.?. We can prove that, for b > 0, v = 1, the function F2 does not belong to the class S.?. Consequently, we obtain that

T?\s£ = 0.

From the detailed considerations it follows that, for b > 0, v = 1 the image of the half-plane n+ in the mapping F2 is the set F2(n+) = {w e C : Re w > a, |w - a| > 2^/6}. In consequence, the function F2 maps the domain n+ onto the set F2(n+) which is not a domain.

We shall prove

Proposition 15. The class S.? is not compact.

Proof. Let us consider the sequence {F”}”£N of functions F”(z) = z + n, z e n+, n =1, 2,... Of course, F” e S.?, n =1, 2,... We shall prove that the sequence {F”}”£N is uniformly divergent on compact subsets of n+ to F(z) = to, with that F e S.? obviously.

Let us take an arbitrary compact set A C n+. Then there exists R > 0 such that A C {z e n+ : |z| < R}. Next, take any M > 0 and let N = [M + R]. Then, for n > N, we have |F”(z)| > |n - |z|| > n - R > M for z e A. From this we obtain the announced assertion.

From the definitions of the classes S.? and Rr and (1.6), (1.7’), (4.1)

we get

Rr C S^.

Let next F = h + g e T?. Denote

F^(z) = h'(z) - g'(z), z e n+.

Then the following proposition is true.

Proposition 16. If F = h + g e T? and

ReF•(z) > 0, z e n+, (4.2)

then f = h - g is a function of the class RR.

Proof. Let F = h + g e TR? satisfy condition (4.2). Then the function

f = h - g belongs (by Proposition 1.4) to the class 7R and satisfies

condition (1.7). In view of Theorem A and the definition of the class Rr, we infer that, indeed, f e Rr.

Let us assume again that F = h + g e 7? satisfies condition (4.2). Let s > 0. Consider the image of a line z = s + iy, y e R, in the mapping F. This is ”a curve” of the equation

w(y) = F(s + iy), y e R. (4.3)

Let next w(y) = Imw(y), y e R. Then we have

w(y)=Im[h(s + iy) - g(s + iy)], y e R,

and

d

w'(y) = dyIm [h(s + iy) - g(s + iy)] = Re [h'(s + iy) - g'(s + iy)], y e R.

Since F satisfies (4.2), w'(y) > 0 for y e R. Moreover, by normalization condition (1.1) for the function F, we get

lim (w(y) - y) = 0.

y—

From the above facts it follows that the function w is a function increasing continuously from -to to +to. Hence we obtain

Proposition 17. Let F = h + g G T^ satisfy condition (4.2). Then, for any s > 0, the line of the equation (4.3) has the property that an arbitrary straight line parallel to the real axis in the (w)-plane has exactly one common point with it. Furthermore, the image of a half-plane Ds =

{z G C : Re z > s} in the mapping F is a connected set lying on the right

of line (4.3) in the (w)-plane.

The last statement follows from the continuity of the function F and from normalization condition (1.1).

Proposition 18. Let {c„}„eN be an arbitrary fixed sequence of real numbers satisfying the condition

tt

En|c„| < 1 (4.4)

n=1

Then the function: i) f of the form

tt

f (z) = z + 1 + E 7—c” , zG n+, (4.5)

w ^ (z + 1)n V '

n=1 V '

belongs to the class TR and is univalent in n+; ii) F of the form

tt

F (z) = z + 1 - E , zG n+ (4.6)

n=1 ' '

belongs to T^. Moreover, F is locally univalent and orientation-preserving in n+.

Proof. Let {c„}„eN be a sequence of real numbers satisfying condition

(4.4). Then the series

tt

g<z> = — E (7+1)7

n=1 V '

is convergent in the set |z + 1| > 1, thus in n+. Consequently, f is

a function holomorphic in n+, whereas F-harmonic in n+. From (4.4)

tt

and (4.5) it follows that Re f'(z) > 1 — £ iz+ij’U > 0 if |z + 1| > 1.

n=1

Hence ([2], p. 88) f is univalent in n+. We also have f(z) = f(z) for

z = z > 0. Therefore Im f (z) has a fixed sign if Im z > 0 or if Im z < 0. Evidently, lim (f (z) — z) = 1, so normalization condition (1.1) holds,

n+3z^TO

and Im f (z) > 0 if Im z > 0 and Im f (z) < 0 if Im z < 0. In this way, i) has been proved.

ii) follows from Proposition 1.5, (4.6) and the fact that lim Re g(z)

n+3z^^

= 0. Furthermore, JF(z) = |h(z)|2 — |#'(z)|2 > 0 for z G n+, where h(z) = z +1, z G n+. This ends the proof.

. 1). The form of function (4.5) comes from the function ^>(Z) = Z + Cn

n=1

holomorphic and univalent in the set |Z | > 1, thus from the function from the area theorem ([2, p. 29-30]). 2) For functions from the class TR, thus from T^, we have no possibility of expanding them in a Laurent series, but we only dispose of normalization condition (1.1) or conditions (1.1),

(1.4). 3) Moreover, from (4.5) and a consequence of the area theorem we obtain |f (z)| < 21z + 1|, z G n+. 4) A special case is the case when cn > 0, n = 1, 2,... Several classes of functions with coefficients of a fixed sign have been investigated in many papers.

Proposition 19. Let a G (0, n), n G N, ck G R, Ak > 0 for k = 1, 2,..., n and

n

E Ak |ck |< 1. (4.7)

k=1

Then the function: i) f of the form

n

fn(z) = z + E cke-Afcz, z G n+ = {z G C : —a < Argz < a}, (4.8)

k=1

is typically-real in n+; ii) Fn of the form

n

Fn(z) = z — Cfce AfcZ, z G п+, (4.9)

k = 1

is typically-real harmonic in the domain n+ (in the sense of definitions of

TR, , modified to n+, respectively); Fn is also locally univalent and

orientation-preserving.

Proof. Let the assumptions of Proposition 4.5 be satisfied. Then, of course, fn of form (4.8) takes real values for z = z > 0. Since |e-Afcz| =

e AfcRe z, therefore if n+ 3 z ^ to, then Re z ^to and lim |e Afcz | =

n+3z^w

0. Consequently, by (4.8), lim (fn(z) — z) = 0. From (4.7) and (4.8)

n+3z^TO

we also have Ref (z) > 1 — £ Ak|ck|e-AfcRez > 0 if z G n+, so ([2], p.

k=1

88) fn is a function univalent in n+ and therefore it is typically-real in this domain. ii) follows from Proposition 1.5 applied to the case of the set n+. Of course, (z) > 0 for z G n+.

. 1) The functions fn — z, n G N, are the nth partial sums of the respective Dirichlet series.

2) In view of the normalization condition, we cannot consider a full Dirichlet series although n+ is a half-plane, i.e. a set of a type of sets of convergence for such series. 3) Also by this condition, we cannot consider fn in n+ but only in an arbitrary fixed ”angle-domain” contained in n+.

Bibliography

[1] Clunie J., Sheil-Small T. Harmonic univalent functions// Ann. Acad. Scien. Fenn. Series A. I. Math. V. 9. (1984). P. 3-25.

[2] Goodman A. W. Univalent functions, t. I. Tampa, Florida: Mariner Publishing Company, Inc. 1983.

[3] Jakubowski Z. J., Lazinska A. On typically-real functions in the halfplane// Folia Sci. Univ. Techn. Resoviensis V. 129. 16. (1994).P. 5-24.

[4] Jakubowski Z. J., Majchrzak W., Skalska K. On complex harmonic typically-real functions// J. Ramanujan Math. Soc. V. 9. 1. (1994). P. 35-48.

[5] Lazinska A. On some classes of holomorphic functions in the half-plane// to appear; On certain subclasses of typically-real functions// Abstracts of Second Intern. Workshop Transform Methods and Special Functions. II. Varna, 1996. P. 37.

[6] Rogosinski W. Uber positive harmonische Entwicklungen und typisch-reelle Potenzreihen// Math. Z. V. 35. (1932). P. 93-121.

[7] Stankiewicz J. Stankiewicz Z. On the classes of functions regular in a halfplane// Bull. Acad. Polon. Sci. V. 39 (1-2). (1991). P. 49-56.