The generalized Koebe function Текст научной статьи по специальности «Математика»

Trudy Petrozavodskogo gosudarstvennogo universiteta

Seria “Matematika” Vypusk 17, 2010

UDK 517

THE GENERALIZED KOEBE FUNCTION

I. NARANIECKA, J. SZYNAL, A. TATARCZAK

We observe that the extremal function for \a3\ within the class Ua (see Starkov [1]) has as well the property that max \ A4\ > 4.15, if a = 2. The problem is equivalent to the global estimate for Meixner-Pollaczek polynomials P3(x; 9).

In [1] Starkov has found max |a3| within the class Ua, which for a = 2 disproved the Campbell-Cima-Pfaltzgraff conjecture, that |a3| < 3 for U2.

The extremal function f0(z) = OOi. Anzn, z G D = {z : |z| < 1} has the form

f (z) =___________________1__________________

J0V ' (1 - zeie )i-iV02—T(1 - ze-ie )i+iya2-T’

with appropriate 0,0 G (_n,n],a > 1,z G D which appears to be very closely connected with Meixner-Pollaczek (M-P) polynomials [2].

For A > 0, x G R, 0 G (0, n) the Meixner-Pollaczek polynomials of the variable x are defined by the generating function

1 OO

Gx(x; 0; z) = —-----e-r———---------- +. = P^(x; 0)zn, z G D.

v ’ ’ ' (1 ze)A—ix(1 ze—.e)A+ix / ^ n v ’ / 7

Therefore, we see that nAn = Pn_i(Va2 _ 1; 0) and the estimate of Pn (x; 0) as the function of 0 G (0,n) is of independent interest and will lead to the bound for |An|. In this note we find sharp bound for |Pn (x;0)|, n =1, 2, 3, which implies that max |a4| > 4.15 for U2, supporting the result of Starkov [1].

© I. Naraniecka, J. Szynal, A. Tatarczak, 2010

Theorem A [2]. (i) The M-P polynomials P„(x; 0) satisfy the three-term recurrence relation:

nP^(x; 0) = 2[xsin 0 + (n _ 1 + A) cos 0]P,^-T(x; 0) _

_(2A + n _ 2)P*_2(x; 0), n > 2.

(ii) The polynomials P„(x; 0) are given by the formula:

P„A(x; 0) = e^e y (A + ix)j(A _ ix)n—j e-2ij'e, n G N U{0}. j!(n _ j)!

(iii) The polynomials P^(x; 0) have the hypergeometric representation

nA / . n\ _ ine (2A)nTT'/' \ i ■ o\-i —2%e\

Pn (x; 0) — e ------j—F (—n, A + ix, 2A; 1 — e ).

n n!

Symbol (a)n denotes the Pochhammer symbol:

(a)n = a(a + 1)...(a + n _ 1), n G N, (a)o = 1,

and F(a, b, c; z) denotes the Gauss Hypergeometric Function.

(iiii) The polynomials y(x) = P-^x; 0) satisfy the following difference equation

eie (A _ ix)y(x + i)+2i[x cos 0 _ (n + A) sin 0]y(x) _ e-ie (A + ix)y(x _ i) = 0.

have the form of Pi

for further calculations:

From Theorem A we have the form of P-(x; 0), n =1, 2, 3, convenient

Po1(x;0) =1 PT(x; 0) = 2(xsin0 + cos 0), (1)

P2T(x; 0) = 3xsin 20 + (2 _ x2) cos 20 + (x2 + 1),

P3T(x; 0) = (x2 + 1)(x sin 0 +

+2 cos 0) + — (x(11 _ x2) sin 30 + 6(1 _ x2) cos 30).

Remark. In our calculations we will use the obvious convenient formula A sin a + B cos a = \JA2 + B2 sin(a + y>),

, A • B

where cos ^ = =, sin ^ =

VA2 + B^ VA2 + B2'

Denote

2x

sin ^0 = , 2 ,, cos 30 = , 2 „, (2)

x2 + 4 x2 + 4

• o 6(1 _ x2)

sin 3i =

cos 3i =

%/x2 + 4%/ x2 + 9%/ x2 + 1 ’

x(11 _ x2 )

%/x2 + 4%/ x2 + 9%/ x2 + 1 ’ x is fixed, and ^(0) = 3 Vx2 + 1 sin(0 + 30) + Vx2 +9 sin(30 + 3i), 0 G [_n, n].

Theorem 1. For the Meixner-Pollaczek polynomials P^x; 0), x > 0, 0 G (0, n) we have the sharp estimates:

|Pii(x; 0)| < 2 Vx2 + 1, |P2(x; 0)| < Vx2 + 1( Vx2 + 1 + Vx2 + 4),

1 V x2 , * v _ , ---------

3 ee[0,n]

|Ps(x; 0)| < - Vx2 + 1 Vx2 +4 max |^(0)| =

= 1 Vx2 + 1 Vx2 +4^3Vx2 + 1 sin(0 + 30) + y^(x2 + 1) sin2(0+30) + 8^ < < Vx2 + 1 Vx2 + 4( Vx2 + 1 + 3 Vx2 + 9), where 0 G (0, n) is the root of the equation

H(0) = cos(30 + 3i) = _ /x2 + 1 cos(0 + 30) V x2 + 9

Remark. Due to the property: ^(n + 0) = _^(0) and H(n + 0) = H(0), the estimates for |P,i(x; 0)|, n = 1, 2, 3 are valid for 0 G [_n; n].

Proof. Using Remark 1, we have for x > 0 :

P>ii(x; 0) = 2 Vx2 + 1 sin(0 + ^i) < 2 Vx2 + 1 with equality for 0i, such that sin(0i + ^i) = 1, where

x1 cos <£>i = , =, sin <£>i = , =.

Vx2 + 1 Vx2 + 1

For P2i ( x; 0) we have

P2(x; 0) = 3x sin 20 + (2 _ x2) cos 20 + (x2 + 1) =

= Vx2 + 1 Vx2 + 4sin(20+^>2)+(x2 + 1) < Vx2 + 1(Vx2 + 4+Vx2 + 1),

with equality for 02, such that sin(202 + ^>2) = 1, where

3x 2 _ x

cos ^2 = , sin ^2 =

2

%/x2 + 1\J x2 + 4’ %/x2 + 1\J x2 + 4

Finally, for Pg ( x; 0) we have

P3(x; 0) = (x2+1)(x sin 0+2cos 0)+-(x(11_x2) sin 30+6(1_x2) cos 30) = = (x2 +1) Vx2 +4sin(0 + 30)+o Vx2 + 1 Vx2 +4Vx2 +9sin(30 + 3i) =

= — Vx2 + 1 Vx2 + 4(3 V x2 + 1 sin(0 + 30) + Vx2 + 9 sin(30 + 3i)) = = — Vx2 + 1 Vx2 +4 • *(0),

where 30 and 3i are given by (2).

In order to find sharp estimate for Po(x; 0) we have to find max №(0)|

3 0<e<n

for fixed x > 0.

The equation ^ (0) = 0 is equivalent to

H (0) cos(30 + 3i) t/x2 + 1 (3)

H(0) = cos(0 + 30) = V , (3)

which is pretty difficult for discussion. However we can restrict ourselves to the case 0 G [0, n], because ^(n + 0) = _^(0) and H(n + 0) = H(0).

Corolary. In the case a = 2 ^ x2 =3, the equation (3) is equivalent

to

2 \X3

cos(0 + 30) + V/3sin(30 + 30) = 0, sin30 = , cos30 = v=

77

or

5t3 + 5V3t _ 7t _ 3V3 = 0, where t = tg 0. (4)

The approximate calculations shows that, the maximal value of ^(0) is given by t = tg0 ~ 0.938. For t = tg 0 ~ 0.938 we obtain max |A4| =

ma^-|P31(x; 0)| > 4.17, which show that for U2, |A4| can be greater than 4.

n

Our result follows simply by taking 0 = — in ^(0). We get

/3 1

A4 = /7(1 +——) sin( — + ) = t. /2(5/3 + 9) > 4.15.

3 4 6

Remark. Another important extremal problem solved by Starkov [3], namely max |argf (z)|, f G Ua, has the extremal function:

fo(z)

1

(e®*2 — e®*1 )*%/a2 — 1

, 1 — ze 1 yVO—1 1

1 — zeit2

t1 = t2 + 2kn,

1 r 1 r

with t1 = n — arctg — — arctg —, t2 = —n + arcsm — — arcsin —,

a a a a

r = |z| < 1, ti = —12.

The coefficients of this function are not M-P polynomials. Inspired by that we are going to study the properties of the generalized Koebe function defined by the formula:

1

(e®^ — e®^)c

and

1- zei0 \ c

kc(0,0; z) = T-^^ —1 ,c G C m e®^ = ei&, z G D

1 1 _

ko(0.*; »)=(e# — e„) '«9 • e‘« = e«, z G D,

for which

kc(0, 0; z) = —--------rj—:-------------------------—-. .. 1 , c G C.

cV ’ (1 — zei0 )1-c(1 — ze^ )1+c

This is evidently connected with the polynomials which we call the generalized M-P polynomials (GMP) given by generating function (0, 0 G R, x G R, A > 0) :

.. OO

°A(x; 0.0; z) = (1 _ ze,<,)A-„(1 _ ze„)A+„ = E (x;0 0)z”. z g D.

This set of polynomials will be studied somewhere else.

Bibliography

[1] Starkov V. V. The estimates of coefficients in locally-univalent family Ua // Vestnik Lenin. Gosud. Univ. 13(1984). P. 48-54. (in Russian).

[2] Koekoek R., Swarttouw R. F. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue // Report 98-17. Delft University of Technology. 1998.

[3] Starkov V. V. Linear-invariant families of functions // Dissertation. Ekatirenburg, 1999. 1-287. (in Russian).

Department of Mathematics,

Faculty of Economics,

Maria Curie-Sklodowska University,

20-031 Lublin, Poland

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