HYPERELLIPTIC INTEGRALS AND SPECIAL FUNCTIONS FOR THE SPATIAL VARIATIONAL PROBLEM Текст научной статьи по специальности «Математика»

84

Probl. Anal. Issues Anal. Vol. 13 (31), No 2, 2024, pp. 84-105

DOI: 10.15393/j3.art.2024.15371

UDC 517.58, 517.54, 517.977

B. E. LEVITSKII, A. S. IGNATENKO

HYPERELLIPTIC INTEGRALS AND SPECIAL FUNCTIONS FOR THE SPATIAL VARIATIONAL

PROBLEM

Abstract. The study of the properties of special functions plays an important role in solving many problems in geometric function theory. We study the properties of hyperelliptic integrals and special functions, which definition includes a parameter that depends on the dimension of the space. The appearance of these functions is associated with the solution of a specific variational problem of finding in n-dimensional Euclidean space a surface that has the smallest area in a given metric among the hypersurfaces formed by rotation around the polar axis of a plane curve connecting two fixed points in the upper half-plane.

Key words: special functions, hyperelliptic integrals, modulus of a family of surfaces, variational problem

2020 Mathematical Subject Classification: 33E20, 30C65, 30C70, 49Q05

1. Introduction. The study of the properties of special functions and their application to solving extremal problems of geometric function theory and, in particular, quasiconformal mappings is the subject of many works by G. D. Anderson, M.K. Vamanamuthy, M. Vuorinen, T. Sugawa, X. Zhang, and others (see, for example, [1]- [4]). Dependence on dimension of the volume of an n-dimensional ball of unit radius expressed in terms of the gamma function, and various relationships associated with this quantity, was studied in [5], [6], [9].

In the work of the authors [8], a solution was obtained to the variational problem; it arose when studying the change in the modulus of a family of surfaces that separate the boundary components of a spherical ring in the n-dimensional Euclidean space En (n ^ 3), upon transition to

© Petrozavodsk State University, 2024

its subfamily consisting of surfaces enveloping the continuum (obstacle) belonging to the ring.

Let x = (xl,x2,... xn) be a point in En, n ^ 3, = Vx\ + x? +... + x2n be the length of x. Let us choose the Oxl-axis as the polar axis in the system of spherical coordinates in En and define a complex structure on the two-dimensional plane Oxlx2, identifying it with the complex plane Cz.

We consider a family of plane piecewise-smooth curves 7 = (9,^) given by the parametric equation z(t) = epPr), T e \9,^\, and connecting the points z0 = rLetd and zl = rle1^ (1 ^ I < L) in the closed set

Br = {z: r ^ |z| ^ Lr, arg z e \9,^\ , 0 < 9 < ^ < n,r > 0}.

It is assumed that at the points of differentiability p'(r) ^ 0 and r) ^ 0. The choice of curve representation is due to the convenience for further analysis.

The variational problem mentioned above is to find, among the surfaces formed by rotation in En around the polar axis of the curves (9,^), the surface of the smallest area calculated in the metric p0(x) = |£|l-n^--l, where un-l is the surface area of the hypersphere unit. This metric is extremal for the modulus of the family of surfaces separating the boundary components of the spherical ring in En [7]. The description of optimal trajectories of the variational problem and the calculation of the areas of minimal surfaces leads to a class of special functions, the representation of which involves hyperelliptic integrals of a standardized forms:

n 1 \ f <fc J? 1 K ; [ ?x2(n-2) -1 dx

a a

(1)

where 1 ^ a < b ^ c <8 and n ^ 3 is a natural number.

The purpose of this work is to study the general properties of such hyperelliptic integrals and apply them to study the extremal properties of the special functions under consideration.

In Section 2, we study the behavior of hyperelliptic integrals (1) depending on the parameters. These results are used in Section 3 to study the properties of special functions that describe optimal trajectories of the variational problem. Section 4 is devoted to the study of extremal properties of special functions related to the area of minimal surfaces formed by rotation around the polar axis of optimal trajectories. The established properties have application to the solution of some variational

and extremal problems for capacities and modules of spatial condensers. In Section 5, we present the results of numerical experiments carried out using PTC Mathcad Prime to construct graphs of special functions and calculate the values of extremal functions and constants defined in the statements proven in the work for various values of dimension.

2. General properties of hyperelliptic integrals 6n and En.

Along with the general representation of functionals (1), we also consider the following particular forms:

On(a, b) = ' dX

V&2 - xWx2pn-2) - 1 '

q — f_^X_

J ?b2 - X2 ?X2pn-2) - 1; 1

^ . 7X . ?x2pn-2) - 1 dx

En{a, ?b2 -X2 ,

^ ,,, , \lx2pn 2) - 1 dx En{b) =

VW-

X2

Note that for n = 3 functionals (2)-(5) are elliptic integrals that can be reduced to the Legendre normal form.

Lemma 1. 1) Qn(a,b, c) is strictly decreases as a function of n and function of a e [1, b), is strictly increases as a function of b e [1; c], and strictly decreases as a function of c e [b, 8) for fixed values of other variables; Qn(a) strictly decreases.

2) For natural n ^ 3 and any values 1 ^ a <b ^ c < 8, the following chains of inequalities hold:

Qn(a ,b, c) ^ Qn(1,b, c) ^ Qn(c) ^ Qn(b) ^ Qn (a) <

K

< An := lim 0n(a) = .--;

a^1 2yjn - 2

On(a,b, c) ^ Qn(a, b) ^ On(b) ^ Qn(a) < An.

3) lim6n(a,b) = lim On(a,b) = 0 and max On(a,b) = On(a,bn(a)),

b^a be(a; 8)

where bn(a) is a solution of equality

(n — 2)x2(-n-2)dx a

Vb2 - x2(x2(n-2) - l)3/^ V&2 - a2^a?pn-2) - 1:

a

or equality

9n(a,b) = ~ '

(n — 2)Vb2 — a2Va2pn-2) — 1 J ^Jb2 — x2\x2p'n-2) — 1\3/2 ' v 7

a

Proof. Direct calculations show that

dQn(a,b,c) 0 SOn(a,b,c) 0 dQn(a,b,c) 0

dn 'da ' db '

dQn(a,b,c) 0 dOn(a,b) 0

dc ' da '

whence follows the strict monotonicity of the function On(a, b, c) in each of the variables, as well as the strict monotonicity of the function On(a, b) in the variable a. Hence, On(a,b,c) ^ On(1,b,c) ^ On(c) and On(a,b,c) ^

^ On(a,b) ^ Qn(b).

Setting x = 1 + (a — 1)y, we replace the variable in the integral

a

On(a) = '

1

■\/a2 - x2Vx2(n-2) - 1'

After simple transformations, we get 1

n f \ f dy

en{a) = 1

0

a + 1 - 2y -(a - 1 Ws£-2) Ck2{n-2)(a - lf-1 y"

From this it follows that d&™0.a) < 0, i.e. Qn(a) strictly decreases, and

_ , , ,. _ , , 1 f dy n

sup &n(a) = Jim 9n(a) = — — = An =

ae(1;«) a^1 " 2y/n - 2 J 1 - y n 2^fn-2'

Similarly, setting x = 1 + (b — 1)y, replacing the variable in the integral On(a, b), we find:

1

Vn(a, b)= f df =. (9)

J a + 1 — 2y —(b— l)y2Vs£ qC2W& — 1)fc-v

q —1 b—1

This implies limfc—a On(a, b) = limfc—8 On(a, b) = 0.

On the other hand, setting x = bsint in the integral On(a, b), we find:

Qn(a, b) = ^ d

y/(b sin i)2(n-2) — 1'

arcsin y

Hence,

B6n(a,b) a / (n — 2)b2n-5 sin2pn-2) tdt

db b^/W—oVa2(n-2) — 1 J [(b sin t)2pn-2q — l]3/2

arcsin y

or, after the reverse replace,

b

dOn(a, b) a n — 2 f x2pn-2)dx

db 6V b2 — aVa2pn-2) — 1 b J ^b2 — x2(x2pn-2) —1)3/2

'10)

Therefore, the maximum value of On(a, b) on the interval b e (a, 8) is achieved at b = bn(a), which is the root of equation (6) or equation (7), obtained from (6) using simple transformations. □

Lemma 2. 1) En(a, b, c) is strictly increases as a function of n, strictly decreases as a function of a e [1, b), strictly increases as a function of be [1, c], and strictly decreases as a function of c e [ b, 8); En(a, b) strictly decreases as a function of a e [1, b) and strictly increases as a function of b e (a, 8) for fixed values of other variables; En(b) strictly increases. Moreover, for natural n ^ 3 and any values 1 ^ a < b ^ c < 8, the following chains of inequalities hold:

En(a,b, c) ^ En(a, b) ^ En(b) ^ En(c) < lim En(c) = 8;

C—>8

En (a, b, c) ^ En(1,b, c) ^ En(c) < 8.

2) The following functional equation holds: d En(b) n — 2

[En(b) + en(b)]. (ii)

dEn(a,b, c) dEn(a,b, c)

Proof. 1) Direct calculations show that---> 0,---< 0,

d n d a

d En(a,b, c) d En(a ,b, c) d En(a, b) .

- ' > 0, -V < 0, and - < 0. Replacing the

d d d a

variable in the integral En(a, b) setting x = b sin t, we find:

En(a, 6)=f X = f y/(b sin i)2(n-2) — 1dt. (12

72

X2

arcsin %

\/b2 — x2

a

Hence,

* /2

3En(a, b) a Va2(n-2) — 1 f (n — 2)b2n-5 sin2pn-2q tdt

+ I / , x ^ 0, (13)

db b V&2 — a2 J V(b sin t)2pn-2) — 1

arcsin y

whence follows the strict monotonicity of functions En(a,b, c), En(a, b) and En(b) in each of the variables, as well as the chain of inequalities given in the formulation of the lemma. 2) From (13), it follows that

*/2

d En (b) n — 2 f (b sin t)2pn-2)

--dt.

db b J y/(b sin t)2pn-2) — 1

arcsin 1

Taking into account the definitions of functions En(b) and On(b), after simple transformations we obtain (11). □

3. Hyperelliptic integrals in the representation of optimal trajectories for the variational problem.

In the metric p0(x) = \x\1-nun-^1, the area F(j) of the surface formed by the rotation in En of the curve j = around the polar axis in

a polar coordinate system, has the form:

7> _

F(^ = ^ f sinn-2 v(t)^(v'(t))2 + (p'(r))2dr. (14)

Wn-1 1 V

The description of the optimal trajectories, providing the minimum value of the functional (14) in the considered class of curves (see [8]), leads to the following functions of the variables 9 and p:

h(9é)=Î sin"'2 9dt ° P J Vsin2p"-2) t — sin2p"-2) 9 ' ( )

if p >9 and sin^ sin^ (p < * or 9 ^ n — p ^ |);

■4>

f sin"'2 pdt . .

hi pd, pq H /. 2(n 2) p 2(n 2 ;, (16)

J Vsin2(ra'2) t — sin2(n'2) p

if p >9 and sinp < sin^ (n — p <9 ^ | or 9 > |);

h (9) = hc(9). (17)

Replacing the variable in the integrals (15)-(17) and assuming

1 1

sin0 sinp sin (n — pq we find:

h° (9 ,pq = Qn(1, 6 sinp, bq, if 0 <9 <p ^ */2; (18)

h°(9,pq = On(bq + On(bsin(n — pq,bq := ~n(b,n — pq = ~n(b,pq, (19)

if 0 <9 — p < */2;

hi (9 ,pq = 8n(1, d sin (n — 9q, dq, if*/2 ^ 9<p <n; (20)

hi (9 ,pq = On (dq + ®n(d sin 9,dq = En(d, eq, if n — p <9 < */2; (21)

h (oq = On(bq, if 0 <9<*/2. (22)

Lemma 3. 1) lim On(1, bsinp, b) = lim On(1, bsinp, b) = 0 and

b^1 /sin

max 6n(1, bsinp, b) = 6n(1, bn(p) sinp, bn(p)),

be(a,<x>)

where bn(p) is a solution of equality

bs i n^

b2dx tanp

(b2 -x2)3{Wx2(n-2) - 1 V(b sinp)2(n-2) - 1'

(23)

2) Sn(b,p) = On(b) + On(bsinp, b) strictly decreases in each of the variables for a fixed value of the other.

Proof. 1) Change of variable in the integral 0n(1, bsinp, b), setting x = 1 + (b sinp — 1)y, leads to the following representation:

6n(1, 6 sinp, b) =

1 _

^J bsinp — 1 dy

bb2 — [1 + (b sinp — 1)y]2bZ2lnr2)C2(n-2)(& sinp — 1)^y) , whence follow the equalities

lim On(1, bsinp,b) = lim On(1, bsinp,b) = 0.

^Vsin

Therefore, for any p e (0,*/2), the maximum value of On(1, bsinp,b) is reached at the point bn(p), which is the solution of the equation

bs i n^

dOn(1, bsinp,b)) tgp f bdx

db by/(b sin p)2(n-2) -1 J (b2 -x2)3/Wx2(n-2) -1

2) It is easy to see that for p e (0,*/2) dEn(b,p) dOn(a, b)

= 0.

dp da

bcosp =--, : < 0.

a-bsin^ ^(b sinP)2Pn-2q - 1

d@n(bsinb) _ B©n(a,i

da

into account equality (10) and equality

+ sind

, then, taking

a=b sin0

dOn(a, b)

-bsine bcosdV(bsind)2(n-2) - 1

1

a

we find

b

dOn (b sin (n — 2) f x2(n—2)dx

db b J ^/tf—x2 (x2(n—2) — l)3/2

< 0.

Because < 0 (see lemma 1), we have < 0. Hence, En(b,^)

strictly decreases as a function of b. □

The properties of hyperelliptic integrals formulated in Lemmas 1-3 imply:

Theorem 1. [8] 1) h0(9,^) is strictly increasing as a function of 9 e (0, n — for fixed ^ e (w/2, n) and

max ho(9, ^) = 2h(n — = 26ra (1/sin J .

de(0 '-k—'^]

2) If 0 < 9 < ^ < * / 2, then lim^q+ ho(9,^>) = lime^—o ho(9,^) = 0 and hn(ip) = maxdep0^q h0(9,^) = h0(9n,ip), where 9 = 9n(ip) is a solution of the equation

* dt tg ^ (24)

cos2 Wsin2pn—2q t — sin2(n—2) 9 Vsin2(ra—2) ^ — sin2(n—2) 9' o

3) h1(9,ip) is strictly decreasing as a function of 9 e [n — n/2) for fixed ^ e /2,rK) and

max hi(9,^) = 2h(n — = 20n (1/sin^ ,

max hi(9, ^) = h(n — = On (1/sin^ .

de[^/2 ^)

4) h(9) is strictly increasing on the interval (0,n/2) and

'K

sup h(9) = lim h(9) = An = --(25)

e /2 2\J n — 2

4. Extremal properties of special functions for the variational problem.

Calculating the areas of surfaces formed by rotation of the optimal trajectories for functional (14) and comparing these areas with the area

of the (n — 1)-dimensional sphere of unit radius calculated in the same metric, leads to the need to study the properties of a number of special functions that can be represented using of hyperelliptic integrals (1). Let us first consider the functions involved in expressing the areas of surfaces formed by the rotation of the optimal trajectories for the functional (10):

" /2

r sin2(ra—2) tdt

H (9) = I Sin tM , (26)

( ) J Vsin2(ra—22)t — sin2(n—2)e

n f sin2(ra—2) tdt

Ho(d " / 2 r 2 2, , (27)

J Vsin2(n—2) t — sin2(n—2) 9 e

if ^ > 0 and sin^ sin^ (0 < 9 < ip ^ | or 9 ^ -k — ^ ^ |);

n } sin2(n—2) tdt , ,

H1(°= / . 2(n 2) . 2fa 2 ; , (28)

J Wsin2(n—2) t — sin2(ra—2) rt e T

if ^ > 0 and sin< sin^ (n — ^ <9 ^ 2 or | ^9 < ip < n).

Replacing the variable in the integrals (26)-(28) and assuming

d - 1 1

sinö' sin—

we find:

b

1 C T2(ra-2)JT 1

H(9) = ^ , -= ^ [En(b) + On(b)]. (29)

V ' hn~2 J Jh2 _ rr.2jrr.2(n-2) _ 1 hn-2 ' V ' x n v '

1

bn-2 J Vb2 - T2^T2(n-2) - 1 b

If 0 <9 ^ f, then

Hq(9,^) = -—^ [En(1, bsin^, b) + On(1, bsin^, b)]. (30) If 0 <9 — ^ < \, then

Ho, = ^ [En(b, n — + ~n(b, n — ^)] , (31)

where

En(b,7 — p) = En(b) + En(bsin(n — p), b) = En(b,p). (32) If 0 <7 — p O ^ 2, then

Hi(°,p) " d-2 [En(d, °) + Sn(d, d)]. (33)

If \ ^ 6 <p <7, then

H1(d,p) = ^ [En(1,dsin(7r — d),d) + Sn(1,dsin(7r — d), d)]. (34) dn 2

Lemma 4. En(6 ,p) is strictly decreasing as a function of p e (0,*/2), strictly increasing as a function of b, and we have a functional equation

dEn(b,p) n— 2 r , _

[En(b,p) + ^n(b,p)]. (35)

B b b

dp

Proof. It is easy to see that < 0 at p e (0, */2). By virtue of (32),

we have ^^ = + (sin p+ ^^) .

V / a=bsinp

Carrying out the necessary calculations and transformations, we find

BEn(a, b)

B a

and (see (13))

BEn(a, b)

a(b sin p)2(n~2) — 1

a-b sinp bcOSp

\a=b sin p

<\J(b sin p)2(n-2) — 1 +

B

b

tgp [77T- _2 7 n — 2 f x2(n-2)dx

b J V b2 — x2Vx2(n-2) — 1'

b sinp

Because

n — 2 f x2(n-2) dx _n — 2

b J ? b2 — x2?x2(n-2) — 1 " b

bsinp

[En(b sinp, b) + On(b sinp, 6)]

taking into account (11), (19), and (32), we get (35), whence it follows

that > □

tT-S

Theorem 2. 1) H(6), Ho(0) = 2H(0)— $ sinn—2tdt and H1(d) = H(0)-

o

n—e

— $ sinu—2tdt are strictly increasing on the interval 9 e (0,71/2) and o

sup H (9) = An, inf H (9) = ^^;

ee(0*/2) Oe(Q,7T/2) 2un—2

sup Ho(d) = 2An — ^^ = A°n, inf Ho (9) = 0; (36)

ee(0*/2) 2un—2 Oe(Q,7T /2)

sup H1(d) " An - ^ " A, inf H1(d) " - ^

2un-2 0e(O,'r /2) 2u„

Oe(Q* /2) 2Un—2 vp(Q,n/2) 2Un—2

2) A° > A^ > 0 for n ^ 3, limra^8 A° = 0, and there are inequalities:

*- < K < ?" 1

2J\Jn - 2 ^ ra^v y/2(n - I)-

*-^ r^< a1 <c!*( ? 1

___^ \

(37)

n - 2 ^ 2\y/2(n - 2) Vn-lJ'

3) The equation

■K-6

Hp9) = f sin™-2 tdt (38)

0

has a unique solution 9 = x(n) 6 (0,^/2)-Proof. 1) Because

* /2

H(9) = f Vsin2(ra-2) t - sin2(ra-2) 9dt + h(9) sinra^2 9, (39)

*/2 "/2

i fVsin2(-2), — sin2(.—2),dt"— f (n — 2) sin2"—5 Qdt ,

de J J Vsin2(ra—2) t — sin2(n—2) 9

e e

then dHdfq = sin""-2 O^^p- > 0, since, according to Theorem 1, ^^^ > 0. Hence, H(Q),H0(9), and H1(9) strictly increase on the interval (0,^/2).

From here, due to (25) and (39), taking into account the equality

V2

I sinn~21dt = , we arrive at (36). o n 2

2) Inequalities (37) is a consequence of inequality

V n — 1 un-2 V n — 2

which was established by K.H. Borgward [6] (see also [9]). It follows that A; > 0 and limra^„ A; = 0, v = 0,1.

3) Since Hl(9) strictly increases on the interval (0,*/2) and takes values of different signs, equation (38) has a unique solution 9 = x(n). □

For v = 0,1 we set:

(9,1) = Hv(9,1) — sin tdt

Theorem 3. 1) H0(9,p) and H0(#,p) are strictly increasing as functions of 9 e (0,n — p] for fixed p e /2,7 — 6], H0(6I,p) takes positive values, and

max Ho(0 ,p) = Ho(n — p).

8e(0,Tr-'4>]

2) H0(6I, p) is strictly increasing as a function ofp e (9,7 — 9] for fixed 9 e (0,*/2), max^e{eH0(0,p) = ^(9), and for any 9 e (0,*/ 2), there exists a unique solution p = (9) of the equation

i>

H0(9= j sinn-2t dt. (40)

0

3) H0(#,p) is strictly increasing as a function of 9 near the point 9 = 0 for fixed p e (9, */2], lim*^0 ^0(0,p) = 0 and

max H0(0,p) = H0(9n,p) > 0,

6e(0,ip]

where 9n = 9n(tp) is a solution of equation (24). In addition, on the interval (9n(1),1) there exists a solution 9 = (1) of equation (40).

Proof. 1) Let 0 <9 ^7 — 1 < 2. By virtue of equality (31), we have

bh pepn - 2)

n— 1

Enpb, 7 -$) + Srapb, 7 -p)

+

1

+

n—2

dEnPb, 7 BEnpb, 7 -p)

db B b

where b = -rL*. Taking Lemma 4 and relation (35) into account, we find

B H (9 ,p) 1 BSn( b,7 — p)

Bb bn 2 Bb

Hence,

< 0.

B Hpo,4>) cos 9 BEnpb,7 0

B d sinn e b b

that is, Ho(9,p) and H0(0,p) are strictly increasing as functions of 9e(0,7 — p] for fixed p e (f,7 — 9]. Since

inf H0 (9,p) = lim H0(0,p) = 0

and by Theorem 2,

max H0(0,p) = H0(7 — ^,^ = ^(7 — p) > 0,

8e(0,Tr-'4>]

then the function H0(9,p) takes positive values for any 9 e (0,7 — p]. 2) Since

BH0 (9 ,p) . n-2 ,( sinn-2p \ -', = sinn 2 m . — 1 > 0,

Bp Vsin2(n-2)rn — srn2(n-2)9 '

for p e (9,7 — 9], then H0(9,p) is strictly increasing as a function of p e (9,7 — 9] for fixed 9 e (0, */2) and

max H0(0 ,p) = H0(9).

Pe(8,TT-8]

Because

inf Ho(0= lim H0pö= - sinn tdt < 0,

pepe ,K—e~\ p^e+o J

0

and by Theorem 2,

max Hop9= ,7 - 9) = Hop9) > 0,

Pe{d,Tr—d]

then, for any 9 e (0,K/2), there exists a unique solution ^ = ^(9) of equation (40).

3) Let 0 < 9 < ^ ^ • . It is easy to see that

BH0( 9 ,j>) _ dH0(9 ,j>) _ . ra_2 Jh0(9, 39 39 Sin 39 '

Therefore, the monotonicity intervals of H0(#, ^) and h0(9, ^) as functions of the variable 9 for a fixed value of ^ coincide. By virtue of Theorem 1 (property 2), this implies that

max Ho(0= Ho(9n

&e(0,p]

where 9n(tp) is a solution of the equation (24).

As H0(0increases near the point 9 = 0 and lim^ciH0(0= 0, H0(dn(ip),^>) > 0. From the definition of H0(0,^>), equality (30), and the properties of the hyperelliptic integrals En(1, bsimp, b) and 6n(1, bsimp, b) (see Lemma 3), it follows that

K — p

lim H0(0= - f sinn—2 tdt< 0. J

0

Therefore, on the interval (9n(tp),^) there exists a solution 9 = 9°n(ift) of equation (40). □

Theorem 4. 1) Hi (0is strictly decreasing as a function of 9 e[n — for fixed ^ e [K/2, n). There are equalities:

max Hi(9,<$) = ^(n — # Hi(K/2^) = HX(n — #

de[K—p,p)

p

inf H1(9,ib) = lim H1(9,ib) = — | sinn—2tdt < 0.

8e[K—p,p) d^'P—0 J

0

2) H1 (9is strictly decreasing as a function of ^ e [K/2,n) for fixed 9 e [n — /2]. There exists a unique solution ^ = ^n(6) of equation

Hi(9 sinn—2tdt (41)

and, besides, pn/2) = n — x(n), where x(n) is a unique solution of the equation (38). In addition, for any p e (n — x(n),n) there exists a unique solution 9 = 9ln(p) of the equation (41).

3) Hi(0 ,p) takes negative value near the ends of the interval p e (9, n) for any fixed 9 e (w/2,p) and

max M1(9,p) = M1(9,pn(9)),

pe(S,Tr)

where p = pn(9) is a solution of the equation

i>

cos21Vsin2(ra-2) t — sin2pn-2q p

1 tg9

42)

sinn-3p cosp Vsin2(ra-2) 9 - sin2(n-2)p'

There exists a solution 9 = 9 (n) of the equation

Hi(0 ,M$)) = 0 (43)

and for 9 e ¡2, 9(n)) there exists two solutions p = pn(9) e (9,pn(9)) and pn(9) e (pn(9),n) of the equation (41).

Proof. The statement 1) follows from the fact that

dHl(9_ sin2(n-2)9

B~9 asin2(n-2)9 - sin2(n-2)p <

for 9 p[k and fixed p e p1,/2,n). It means that H1(6I,p) is strictly

decreasing as a function of 9. 2) As

i>

Hi{9 ,p)=( \Jsin2pn-2qt — sin2pn-2q pdt + sinn-2phi{9

BHld" sinn 2pdhld4>'^ • By virtue of (20) and Lemma 3 (property 2), dhi(9,p) BEn(d, 9) 3En(d, 9) cos(n — p)

Bp Bp B d sin2 (n — p)

It follows that

B5i<m " ^BMLil _ A < 0.

Oty \ Oty /

It means that ,n — 1) is strictly decreasing as a function of 1 for fixed 9 e \n — 1,k/2]. Therefore, by virtue of Theorem 3,

max rn1(d= rn1(d,n — d) = H0(d)> 0,

'e[K—B,K)

inf H1 (0,1) = lim H1 (0,1) = — f sinn—2 tdt< 0.

'E[K—6,K) '^K j

0

It follows that there exists a unique solution p = (9) of the equation (41). Since Hi(K/2,1) = H(n — 1),n — ^(K/2) = x(n).

By Theorem 2, H0(n — p) and H1(n — p) are strictly decreasing on the interval 1 e (K/2,n). Therefore,

inf M1(d,1) = H1(n — 1)< 0,

de[K—'^ /2)

max ,1) = Ho(n — 1) > 0

9E[K/2)

for 1 e (n — x(n),n).

Consequently, for any 1 e (n — x(n),n) there exists a unique solution 9 = 9\ (1) of equation (41).

3) Let 9 e (K/2,1). In this case,

H1(0,1) = Ho(n — — — j sinn—2 tdt.

0

Therefore,

lim ,1) = lim ,1) = — sinn-2tdt < 0.

'^K J

0

As Ho(n — — d) =

K-e

= J \jsin2pn-2)t — sin2pn-2) 1dt + sinn-2(n — 1)ho(n — — 9), then

k—'

dho(n — —

Therefore, max'e(0,K) H1(61,1) = H1(61,1n(9)), where 1 = 1n(9) is a solution of equation dh°(K—',K—d') = 1. Using representation (18) and repeating the calculations performed in the proof of Lemma 1, this equation can be represented in explicit form (42). Because

max H1(0,1) = H1(K/2,1) = HAn — 1)

ee[* /2')

and by Theorem 2

sup Hi(tt — Ip) = A* > 0;

12,-K)

then H1(0,1) > 0 for values 9 and 1 close to K/2 and H1 (9,1n(9)) > 0 for 9 close to K/2. Therefore, there exist a solution 9 = 9(n) of equation (43) and for 9 e (K/2, 9(n)) there exists two solutions 1 = 1n(9) e (9,1n(9)) and 1n(9) e (1n(9),n) of the equation (41). □ 5. Results of numerical experiments.

Let us present the results of numerical experiments on constructing plots of the functions under study.

Figures 1 and 2 present the results of numerical experiments on constructing plots of the functions 63 (a, b) and 65 (a, b) for specific values

^ (?23 , ? , 2

ft'")

<¡>3 (2,6)

Figure 1: The plots of 63(a, b)

Figure 2: The plots of 65(a, b)

1

Figures 3 and 4 present the results of numerical experiments on constructing plots of the functions h0(9,1) for specific values 1 e (K/12,K/6,K/ 4,

n/3,5w/12), when n = 3 (see Figure 3) and n = 5 (see Figure 4). These plots suggest that 9n(p) is a unique solution to the equation (23) for every P e(0, */2).

Figure 3: The plots of Figure 4: The plots of h0(9,p),n = 3 h0(9,p),n = 5

Figures 5 and 6 present results of numerical experiments on construct' 'K ^ ^ ^

• 12, 6 , 4 , 3 , 12 -

ing plots of the functions Ho(0 ,p) for specific values p e (12, | , 4 , 3 , 5f),

when n = 3 (see Figure 5) and n = 5 (see Figure 6).

Figure 5: The plots of Figure 6: The plots of Uo(0,p),n = 3 Uo(0,p),n = 5

These plots suggest that (p) is a unique solution to the equation (39) for every p e (0,*/2).

Figures 7 and 8 present results of numerical experiments on constructing plots of the functions H1(0, p) for specific values 9 e (Q(n)— n/200 ,d(n), 9(n) + */200), when n = 3,9(3) « 1.6435 (see Figure 7) and for n = 5,9(5) « 1.5946 (see Figure 8).

Figure 7: The plots of Figure 8: The plots of rn1(d,p),n = 3 rn1(d,p),n = 5

Let us present the results of numerical experiments on studying the dependence on the dimension n p (3, 4, 5, 7) of the constants defined in Lemma 1 and Theorem 1 (Table 1), as well as the values of extremal functions 9n, , Q\ (Table 2), functions , (Table 3), function (Table 4), and functions pn(0) and pn(0) (Table 5) in specific points.

Table 1: The values of certain constants.

n Ara A0 n A1 n x(n) d{n)

3 1.571 2.142 0.571 1.073 1.6425

4 1.111 1.436 0.325 1.272 1.6057

5 0.907 1.147 0.240 1.347 1.5946

7 0.702 0.872 0.169 1.411 1.5862

Table 2: The values of extremal functions 9n, , .

n on (f) M4) D ^ (4) % -

3 0.628 0.452 0.914 0.667 1.237 1.001

4 0.691 0.503 0.893 0.652 1.184 0.915

5 0.733 0.538 0.892 0.653 1.153 0.870

7 0.786 0.582 0.901 0.663 1.115 0.824

It is of interest to obtain explicit estimates of the extremal functions and extremal values defined in the work, depending on the dimension.

The authors would like to thank the anonymous reviewer for the attention to the work and comments that allowed us to correct shortcomings and improve the manuscript.

Table 3: The values of extremal functions ^ and ^

n rn (!) < (!) rn (! - 0.1) tâ (!) rt (!) < (2 - o.i)

3 0.914 1.176 1.512 2.749 2.572 2.204

4 0.9332 1.199 1.521 2.601 2.404 2.020

5 0.9333 1.202 1.524 2.533 2.330 1.947

7 0.922 1.196 1.526 2.467 2.256 1.880

Table 4: The values of extremal functions

n + o.i) U! + 9JP) 0{n)) ^ (2!) {t - 0.1)

3 1.868 1.753 1.825 2.294 3.057

4 1.827 1.680 1.722 2.235 3.052

5 1.805 1.652 1.684 2.205 3.050

7 1.780 1.628 1.652 2.173 3.047

Table 5: The values of extremal functions (9) and ^n(9).

n tn (2 + 0.01) U ! + ^ ) tn (! + 0.01) U ! + ^ )

3 1.868 1.753 1.825 2.294

4 1.827 1.680 1.722 2.235

5 1.805 1.652 1.684 2.205

7 1.780 1.628 1.652 2.173

References

[1] Anderson G. D., Vamanamurthy M. K., Vuorinen M. K. Topics in special functions. Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday, Report Univ. Jyv"askyl"a. 2001, vol. 83, pp. 5-26. DOI: https://doi.org/10.48550/arXiv.0712.3856

[2] Anderson G. D., Vamanamurthy M. K., Vuorinen M. K. Topics in special functions. II. Conformal geometry and dynamics, 2007, vol. 11, pp. 250-270. DOI: https://doi.org/10.1090/S1088-4173-07-00168-3

[3] Anderson G. D., Sugawa T., Vamanamurthy M. K., Vuorinen M. K. Special functions and hyperbolic metric. Manuscript, 2007.

[4] Anderson G. D., Vuorinen M. K., Zhang X. Topics in special functions. III. In Analytic number theory, approximation theory, and special functions, Springer, New York, 2014, pp. 297-345.

DOI: https://doi.org/10.1007/978-1-4939-0258-3_11

[5] Bhayo B. A. On the inequalities for the volume of the unit ball Qn in Ra Probl. Anal. Issues Anal., 2015, vol. 4 (22), Issue 2, pp. 12-22.

DOI: https://doi.org/10.15393/j3.art.2015.2849

[6] Borgwardt K. H. The Simplex Method. Springer, Berlin , 1987.

[7] Caraman P. n-dimensional quasiconformal (QCf) mappings. Editura Acad. Romane, Bucharest; Abacus Press, Tunbridge Wells; Kent, 1974.

[8] Ignatenko A. S., Levitskii B. E. Method of the Optimal Control in the Solution of a Variational Probem. Science Journal of VolSU. Mathematics. Physics, 2016, vol. 6 (37), pp. 28-39. (in Russian)

DOI: https://doi.org/10.15688/jvolsu.2016.6.3

[9] Zhang H. New bounds and asymptotic expansions for the volume of the unit ball in Rn based on Pad'e approximation.. Results Math., 2022, vol. 77, 116, pp. 1-15. DOI: https://doi.org/10.1007/s00025-022-01652-1

Received December 12, 2023. In revised form, March 21 , 2024. Accepted May 02, 2024. Published online May 11, 2024.

B. E. Levitskii Kuban State University

149 Stavropolskaya st., Krasnodar 350040, Russia E-mail: bel@kubsu.ru

A. S. Ignatenko Kuban State University

149 Stavropolskaya st., Krasnodar 350040, Russia E-mail: alexandr.ignatenko@gmail.com