On extremal problems in the classes of functions with positive real part and typically real ones, II

奈良教育大学学術リポジトリNEAR

On extremal problems in the classes of functions with positive real part and typically real ones, II

著者 SAKAGUCHI Koichi

journal or

publication title

奈良教育大学紀要. 自然科学

volume 18

number 2

page range 1‑6

year 1969‑11‑29

URL http://hdl.handle.net/10105/3094

?&&&«*SB?,® S1SS= S!!# fa«ft-

BullVNara'U. lSduc.rVoI. 18\~ No.X"(Nat.)^ 1969

On extremal problems in the classes of functions with positive real part and typically real ones, H

K6ichi SAKAGUCHI

(Department of Mathematics, Nara University of Education, Nara, Japan) (Received May 31 , 1969)

In this paper the author will show two examples of applications of the fundamental theorems established in the preceding paper [1].

1. Application (1)

We denote by P the class of analytic functions p(z) with positive real part in | z]<O and normalized so that ^>(0)=1. It is well-known that if p(z)-l+a.iz

-\- is a function of P, then the value of p(jz) at z, [2|<O, lies on the closed disc with centre (1+|z[2)/(l-|z|2) and radius 2|z[/(I-|zI2).

Applying Theorem 1 of the preceding paper [1] , we can determine more in detail the variability-region of p(z) in terms of z and the first coefficient a-i as follows. We may assume without loss of generality a^ to be real.

THEOREM3. Let p(z)^l+a1z+--- be a function of P, and let ^=2/5,

-lfSl/35^1. Then p(z), |z|<O, ^es on the closed disc with centre (1.1)

and radius (1.2)

O-l*I8)Q--1*!'+£*-£*)+2H8Q-£2) (l-iz|2)(l+|z|2-/3*-/3i)

2| z|2(l-/32) (l-|z|2)a+UI2-/32-/3i)'

Moreover p(_z) lies on the boundary of this disc if and only if pi_z) is a func- tion of the form

or, equivalently, /3+z

£|=1,

(1.4) Pa(z)= 2z(l -/32) /32- (l-\/l-/32)2f

>

-z+z\/l- -<j3z-i+s/i-e*yଠ'

1 3-z $-z

£1=1.

PROOF. When />(«)=l+axz+•E-•E belongs to P, by virtue of Theorem 1 of

the preceding paper [1] the variability-region of the point (alf p(z)~) in the two-

Koichi SAKAGUCHI

dimensional unitary space forms a closed convex domain A whose boundary is given by and only by the functions of the form

(1.5) where (1.6)

Piz)=7i \1£à¬\ZZ +72 |*f|* =l+2(y1e1+y2s2)z+

=1, £it^2, 7i^0, 72^0, 7i+72=l.

Denoting by D the variability-region of pQz) considered in this theorem, D ap- pears when A is cut by the plane ai=2/3. Therefore D is also closed and convex, and its boundary is given by and only by the functions of the form

(1.5) which satisfy the conditions (1.6) and

(1.7) 71£1+72£2=/5.

From (1.7), 71+72=1, and Ex=££t) we have

(1.8) 7i= £z-(3

72= £2-al

=I£21-1, we have

Moreover from the fact that 7j is real and |£]

(1.9) £2(l-0O-£+£i=O.

When -1</3<1, because of 1-/3^^0 we have £2=(/S-£1)/(l-i8à¬1), which

is not equal to £1. When £=±1, (1.9) becomes (IT^XlT^^O, and so

£i=±l or £2=±1. Therefore from (1.9) and \S1\=\62\=l, Sx^e2t it follows that

(1.10) _<~9.-•E _ _' /3-£i

l-/9£i -1</5<1, |£i|=l or

(1.ll) e2^à¬1=/3=±l, |£2|=1, or £^^=£=±1, 1^1=1.

Conversely, if one of (1.10) and (1.ll) holds together with (1.8), then as is easily verified both (1.6) and (1.7) hold, and then the function (1.5) becomes

a-£i*Xl -/3£i +#2-£iz) +2£i*a-£2) a-£i*Xl- 0£i-#K+£12) or

*O)=- «i =1,

according to (1.10) or (1.ll). The former function is of the same form as (1.3), and if we put /3=±1 in (1.3), Pi(z) reduces to the latter function, namely (l±;z)/(l+2). Consequently the boundary of D is given by and only by the functions of the form (1.3) for every /3 such that -1^/3^1.

To show that the value of PxQz) at a fixed z moves on a circle as à¬varies

under the condition that |£|=1, we rewrite pi(z) in the form

On extremal problems in the classes of functions with positive real part and typically real ones, H

2z(l -/?2) 1 -/36

/3-z (l-£z)(l+£*-/9z-/S£) and consider the transformation

f_/ve+i-\/i-/32

/3+ (l-v/l-/32)\/£

which maps the circle |£|=1 onto |^|=1. pi(z) is then transformed to the function (1.4). Since (1.4) is a linear fraction in £', PzQz) describes a circle as £ varies on |f|=1. This circle is nothing but the boundary of D. From (1.4) we can find by a brief calculation (1.1) and (1.2) as its centre and radius.

REMARK. This theorem will be able to be derived also from an iterated form of Schwarz's Lemma given by Finkelstein [2].

COROLLARY 1. With the hypothesis of the theorem, zve have for |2|<O

(1.12) 1-

1+ -^' /K ^ReKOS?

f *- I*-""

(l+izl2+2l/3zl)(l+H2-2|/3zl)

a-lzpoa+i*!2-/^-/^)

Fach equality sign appears only for one of the tzvo functions of the form (1.4) zvhich are determined by

(1.13) f=-expC2/arg{2(^-l+v/l-/3z)2-z(/3-2+zv/l-/S2)2}]

and

(1.14) f=expC2mrg{2(^2-l+v/l-/32)2+2(ye-^+^v/l-^2)2}3.

PROOF. Since ftz-@z is purely imaginary, we have at once (1,12) from (1.1) and (1.2). Next by solving the equation

d

do -Re/>2GO=0, f=e"

we easily obtain (1.13) and (1.14). Therefore Re/>2(z) attains to its minimum for one of fs given by (1.13) and (1.14), and to its maximum for the other.

COROLLARY 2. With the hypothesis of the theorem, zve have for | z|<O

(1.15) l-U

1+2 /3z 4- ^<RepQz^ <~ ^{1+2 (3z+}

1-he

The left-hand equality sign appears only for the function (1.16)

at z=-/3|z|/| /3j, awe/ Me right-hand one appears only for the function

(1.17) pM=A+l^_

at z=0\z\/\/3

PROOF. When |z| is fixed, the left and right-hand sides of (1.12) attain to

Kdichi Sakaguchi

/|/S| and z=@\z\/\(3 /[/3| and z=/3|z|/|

their minimum and maximumrespectively for z= -, Hence (1.15) follows. Moreover for both z=-/z

(1.13) and (1.14) become £=-1 and f=l respectively, and then by a brief calculation we see that P2(.z) reduces to (1.16) or (1.17) according to £=-1 or 1. Therefore the assertion concerning equality signs is true by Corollary 1.

2. Application (2)

We denote by T the class of analytic functions t(j£) typically real in |z|<Cl and normalized so that z(0)=0, ^(0)=l. Jenkins [3] has proved, making use of an integral representation for the functions of T due to Robertson [4], the following:

THEOREMA. Let t(z)=z+a2z2-\ be a function of T, and let a2-2/3,

-1^/3^1. Then for 0<r<l

l-2/3r+r2 <*0)^ ¹⁺⁰ _(1-r)2 ^1-/3 1-r2 </r^1+/8 1+r (l-2/5r+r2)2- W- 2 (1-r)3+

a+ry

1 -/3 \-r

(1+r)3 ' The left-hand equality signs appear only for the function

r . z

*iW- 1_2/3z+zz '

and the right-hand ones appear only for the function h(z)=- 1+/9

+å 1-/3

( I"*)2 ^{(1+z)2 å}

This theorem will be derived directly, if we can determine the variability- region of the point (Xr) , ^(r)) in the two-dimensional Euclidean space for t(z) satisfying a2=2fi. In fact we have

THEOREM4. With the hypothesis of Theorem A, the point (t(r), *'(>)), 0<Cr<Cl, lies on the closed convex domain bounded by the two curves

(2.1)

and

(2.2)

u + B r

x ‑  1 + u ( 1 ‑ r ) 2 u + /3 1 + r y = ' a ‑ r y

/3 r

x = l + ｫ ( l + r ) : u ‑ B 1 ‑ r

-h 1-/3

+

1 + M l + 2 u r + r 2 1 ‑ /3 1 ‑ r 2

1 + M

1 + /5

( l + 2 ｫ r + r 2) 2

r

1 + M l ‑ 2 u r + r 2  '

1 + /? 1 ‑ r 2

(u : parameter)

\ ^y- n

v ^{1 4--} M - 97vr-4-*-2>l2

^ X i « V-L i ' J x i m. VX w>" i ' y

When /3=1 or ^1, this closed domain reduces to one point

On extremal problems in the classes of functions with positive real part and typically real ones, II 5

/ r 1+r \ / r 1-r \

\(1-r)2' (1-r)3)or \(1+r)2' (1+r)3 /

respectively. When /3=t^±1, (2.1) *'s a cwrf^ strictly monotone-increasing in x, convex upwards, and connecting the tzvo points

( \ 1-2/Sr+r2 ^{. t} ' (l-2£r+r2)2)> ^{1-r2 \}

1 +/3

I"/3 1+/3 1+r , 1-/3 1-r

TXa -f" -\

(1-r)2^ 2 (1+r)2' 2 (l-r)3"7" 2 (1+r)3/

and (2.2) is a curve strictly monotone-increasing in x, convex doxvnvoards, and connecting the same two points. Moreover the point (^(r), *'(r)) lies on (2.1) if and only if t(z) is a function of the form

(2.3) *0>=-T+7- (I-*)2 +^T^T 1+2^+z2 > -^«^L

tm<i /zes ow (2.2) if and only if t(jz) is a function of the form

\+u (1+2;)^ 1+w 1-Zuz+z2 ' ^- -

PROOF. When t(z)=zJra2z2j\ belongs to T, by virtue of Theorem 2 of

the preceding paper [1] the variability-region of the point (a2, t(r), t'(r)') in the three-dimensional Euclidean space forms a closed convex domain A' whose boundary is given by and only by the functions of the form

(2.5) Kz)=7i (1^)2-+72 i+2uz+z2 =2:+2(±fyi-^2>2+--->

where

(2.6) -l^w^l, 7i^0, 72^0, 7i+72=l.

The variability-region of (*(r), /(r)) considered in this theorem appears as i' is cut by the plane a2-2P. Therefore it is also closed and convex, and its boundary is given by and only by the functions of the form (2.5) which satisfy the conditions (2.6) and

(2.7) ±7i-«73=#. '

By finding the value of yt and <y2 from (2.6) and (2.7), we see that such functions are of the form (2.3) or (2.4). The set of the points (*(r), *'(r)) which appear for the functions represented by (2.3) or (2.4) clearly forms the curve (2.1) or (2.2) respectively.

Next, for the curve (2.1) we have dy (l+r) {3(l-r)2+2r(H-«)}

dx r(l-r) {(l-r)2+2r(l+*0}

d2y (l+r)(l-r)3 ^0

dx2 ~ (l-/5)r3 ^

>o,

6 Kdichi Sakaguchi

Therefore this curve behaves as stated in the theorem. Similarly we can observe that the curve (2.2) also behaves as stated in the theorem. We thus complete the proof.

It is clear that Theorem A can be derived immediately as a corollary from this theorem.

References

[1 ] K. Sakaguchi, On extremal problems in the classes of functions with positive real part and typically real ones, I, Bull. Nara Univ. of Educ, 17 (1969), 1-12.

[2] M. Finkelstein, Growth estimates of convex functions, Proc. Amer. Math. Soc, 18

(1967), 412-418.

(_3] J. A. Jenkins, Some problems for typically real functions, Canad. J. Math., 13

(1961), 299-304.

[4] M. S. Robertson, On the coefficients of a typically-real function, Bull. Amer. Math.

Soc, 41 (1935). 565-572.