ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X Y II (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I : PRACE MATEMATYCZNE X Y II (1974)
J erzy K aczmarski (Lôdz)
On the radius oî convexity îor certain regular functions
1. Let a, 0 < a < 1 be arbitrary fixed number and denote by Ü the family of functions oo(z), co(0) = 0, regular in the disc К = {z: \z\ < 1}
satysfying the condition \co{z)\ < 1 for every z e K . Next let p (a ) be the family of functions
(1.1) Р(г) = 1 + асу ( 2 ),
where co(z)e Q and ze K . Evidently p (a ) <=. p , where p is the family o f all functions
p(z) = l - b M + M a + . ..
regular in K, such that
rep(z) > 0 for every z e K . Moreover, denote by 7(a) the family of functions
(1-2) f(z) = z + a 2z*+ ...
regular in К and satysfying the condition /'(^)ep(a) for ZeK .
Evidently 7(a) is a subclass of all functions (1.2) whose first deriv
ative has positive real part in K. Then, every function of 7(a) is schlicht in К [3].
In this paper the exact value of the radius of convexity for 7(a) is given.
R em ark. The definition of 7(a) is equivalent to the following definition. Let f(z) satisfy (1.2) and be regular in K\ f(z)e 7(a) if and only if
\f'(z) — l \ < a for ZeK .
2. It is easy to prove that the family 7(a) is compact. Hence the
radius of convexity of 7(a) is equal to the smallest root rQ7 0 < r 0 < 1,
374 J. K a c z m a rs k i
of the equation g (г) = 0, where
Г z f" (z )l g(r) = min re 1 + •
[г|=Г<1 L J \Z) J
№tV{a)
If f (z )e V (a ), then f'{z) = P { z) for some function P{z) in p (a ).
Therefore
zf"(z) „ , *P'(z) 1 +
/'(*) = 1
-P(s)
and
g(r) = min re
|z|*=r<l
P ( z ) e p ( a )
L Р(г) J
It is well known [2] that p (z )e p if and only if 1 -f- со (z)
(2.1) p(z) =
1 —
G)(z)
for some co(z)e Q. Hence, because of (1.1), P { z)cp (a) if and only if [ l + a)p(z) + l — a
( 2 . 2 ) P (z) =
p{z) + ± for some p{z)ep.
Let zQ be arbitrary fixed point of K . It is known that every boundary function with respect to the functional p{z0), р(я)ер, is of the form
p M 1 pEZ
lei = 1, 1 — 8Z’
hence every boundary function for P (z 0), P (z )e p (a ) is of the form
(2.3) P Q{ z ) = l + aez.
It follows immediately from (2.3) that the set of value of P{z0), P { z )e p (a ) is the closed disc K (c,
q)with the centre c and the radius
q,where
(2.4) c = 1, q = ar, \z0\ = r .
Denote by p 2(a) the subclass of p{a) which contains all functions (2.2), where
(2.6) p(z) = ^ r ^ P i ( 0 ) + ^ r ^ P 2(0), - 1 < Я < 1 ,
P M =
2
л ± ' '2 1 + SfcZ
-, ?
l ~ £ k Z
( 2 . 6 ) Ы =1,A? = 1 , 2 .
Badius of convexity
375
Л ext, consider the function ^ ( u , v) analytic in the semiplane те и > 0 and in the plane v, such that
\K l2+\K\2 > o at every point (u, v).
Since every boundary function of p with respect to the functional
^ [p (z ), zp'(z)), \z\ = r < 1, is of the form (2.5), [4], then every boundary function for #"(P( 2 ), zP'(z)) in p (a ), \z\ = r < 1, belongs to p 2(a). Hence
Г sP'(s)l
(2.7) g(r) = nun re 1 + — — .
\ z \ = r < l
L * \ z) J
P ( z ) c p 2 {a)
L emma 1. I f P (z )e p 2(a), then for z = гег,р, 0 < r < 1, 0 < <p < 2 tc
we have
(
2
.8
)where (2.9)
(
2
.10
)P(«) = 1 + t y ,
h = q 2iZ — x
1 So. e№ 2 z — x
\2z — x\
xz — 2
x = (1 + Я)е2 + (1 — ^)ei5
q is given by (2.4) and 0 < h < q . P roo f. Let P (s)ep 2(a); then
\xz — 2 1 жг —2
P(z) ( 1 + Л) [(1 + a ) p M + 1 ~ a] + (1 ~ Я) [(1 + a)p2 (*) + 1 - a]
(1 + AJCPiO?) + 1 ] + (1 ■- A) [p2(*) + 1]
for some functions p k{z) of the form (2.6).
Therefore P (z) can be represented as follows
(2.11) P (* ) = (1 + X )-P 1(z)Q1(z) + ( l - X ) P t { z ) - Q %{z), where
(2.12) = 1 + < а д Q M = for fc = l, 2.
A — XZ
Since
(2-13) (1 + АШ *) + (1-,Я Ш г) = 1 ,
then
P (z) = 1 -f ae1’ e2z 2 z — x xz — 2 Hence, Р{гещ) is of the form (2.8), where
l y — 1 e<>.e№ 2 гег<р — x
xtrei(p- 2 6-
(2.14)
376 J. K a c z m a rs k i Because of (2.14)
2гещ — х
thus
h2
le2 = Q2
^ |2 — хгегч>\2 xre%4> — 2 ( l — r 2)(4 — \oo\2)
Since
4 - |a?[2 = 4(1 — A2)sin2 •- - У- ? yk = arge*, f c = l , 2 . then
(2.16) Te2
=4 1 - 4
( 1 - Я 2) ( 1 ~ г 2 } У г - у ,
\2 — xret,pf sur Hence Te < q .
L emma 2. I f P (z )e p 2{a), then on \z\ = r < 1
<2.16) gP'(g) = f ( s ) - i - g3~ n (г)гГ 1|а V, a ( l —r2)
P ro o f. Differentiating function (2.11) with respect to гг, we obtain because of (2.11)-(2.13)
zP'(z) = [ _ P ( z ) - l ] + [ ( l + A)P1(z)Q'l (z)z + ( l - X ) P i m ' t ^ ' i - Since
x (1 —* ) ( « ! - e8) 4 (l + A)(ea- e i )
then the second term of the last sum may be written in the form r az2 (1 — А2) (ег — e2)2
Therefore
zP'(z) = (P (z) — l ) -j (2 — xz)2
a (1 — А2) (e1 — e2)2z2 (2 — xz)2 The second term of the last sum may be represented as
* 2 a(l —A2) . yx — y 2
4т] 2Г—:--- — sin where
(2.17)
|2 — xz\2 " 2
2 — xz
V = el ' S2- 2 — xz
Eadius of convexity
377
From (2.15) and because of q = ar we have
« ( l - ^ 2) . я V i - 7 2 q 2- №
_____________ _ o in a --- --- #
[2 — xreÎ9\2 2 4ar2(l — r 2) Hence, for z = rei<p, zP '(z) is of the form (2.16), where
(2.18) r) = e2ivr}*.
3. T
heorem. Let
f(z) = z-\-a2z2-\- ...
be the regular function for \z\ < 1 belonging to the fam ily V (a), 0 < a < 1.
Then f(z) is convex in \z\ < r, ifO < a < a0, and f(z) is convex in \z\ < f, if a о < a < 1, where
(3.1)
(3.2)
л / 21^1 — a
^ Г V l — a + V l + 3(
r = f (a) = 2a and
(3.3)
The function
(3.4) f ( z )
i + V &
ei<p
— /o(e“’vz), 0 < ç) < 2тг,
°/ P («)? 0 < a < a0, where
(3-5) jfo(«) = a i ^ ^ l o g ( l - f e ) + |i + a i - ^ b + | -
2:2,
(3.6)
and the function
2 r 2 —1
у 3
(3.7) /**(*) = 0 < 9 ? < 2 тг,
lo g l = О,
°/ F(a), a0 < a < 1 sTiow that r and r cannot be replaced by larger numbers.
For a = 1 we obtain r = \ [1].
P ro o f. 1) Let P ( z ) e p 2(a). Then because of (2.2), (2.5) and (2.6), where ek = ег?к, and in view of Lemma 2, g (r) given by (2.7) can be repre
sented for z = гег<р, 0 < r < 1, 0 < cp < 271 in the form
(3-3) g(r) = min теП{Р(гег9>)),
378 J. K a c z m a r s k i where
тг/ . . a 2r 2 — \w — 112 H(w) = 2 — w 1--- ri.
a { l - r 2)w n
- 1 < / . < 1 , 0 < < 2
k, 0 < < ? < 2 tc (Je = 1 ,2 ).
Let (3.9) Then
Р (гег<р) = selt, s > 0 , im if = 0 .
g(r) = min ire s,i [ -
it
a V -| s e “ - l |2 „
a (l — r 2)s ]
Since re 97 e г* < 1 ,
where
gr(r) > шщФ(«, t),
_
1
Г1 —a2?*2 / a (l —r 2)\
"1Ф(«, *) = —73--- - s + 2a(l —r*H--- ( 2 H --- ' cost
a (l î"2) L ® \ s / J
The function Ф($, t) is defined in the region
D = {(s, t): l — a r < s < l + ar, — y>(s) < t < y(s)}
and on its boundary dD, where
s2 + 1 — a 2 r 2
(3.10) ip(s) = arc cos
2s , 0 < v(*)< V(e0), with $ 0 = \Vl — a 2r 2\(x).
If, at some point ( s * ,f ) of the region D, 0 ( s * , f ) = min Ф (s, t),
(s,t)W
then s*, t* are the solution of the system of equations дФ(«, t)
= 0. d 0 (s, t)
ds dt
with the unknowns s and t, i. e. of the system
1 — a 2 r 2 a ( l — r 2)
= 0
+ cos if = 0 , [2 + - “h y i l ] ■ sint = 0 . Because of (3.10) this system has the solution:
(3.11) = 8x(r) = V'tl —a )(l + <z r 2), tx = 0.
(!) In the sequel \Va\ for a > 0 will be denoted by Va.
Radius of convexity
379
Since
--- aii---> ° and — --- №--- ( Э.Й ) > 0 ’ we have t* = #x and s* = s1? if 5xe I, where I = {s: 1 — ar < s < 1 + ar}.
It is easy to verify that s1e I if and only if r > r*, where (3.12) r* = r*(a) = [1 + ^2(1 - a ) ] ' 1.
Thus
min 0 {s, t) = — — —
(S, t)eD
a ( l — r 2) [V(1 - a) (1 + ar2) + a(l - r 2) - 1]
for r > r*.
If (s, t)e dD, then y>{в)) = x (s), where 1 — a2r 2 with 1 — ar < s < 1 + ar.
Therefore
X(s) = H 3
for 0 < r < 1.
Let
(3.13) and
1 — 2ar min Ф(в, t) = %(1 — ar) = —---
(s,t)edD
1 — ar
h (r) = min Ф{в, t), (s,t)eD^dD s 2 = s2{r) = 1 — ar
T(e) = Ф(«, 0) for 1 - ar < s < 1 + ar, r > r*.
We have
T (s2) - T ( Sl) = + +
where 0 < ê < 1.
Since T'^j) = 0 and T" {s) > 0 for every s > 0, then T^j) < T{s2) for r > r*.
Therefore
Л (r) =
2 [У(1 - a) (1 + a r 2) + a ( l ~ r 2) ~ 1 ] a ( l —r 2)
1 — 2 a r
for r ^ r*j
1 —ar for 0 < r < r*.
380 J. K a c z m a r s k i
If r > r*, then the equation h (г) = 0 is equivalent to (3.14) h(r) = ar4 + ( l — a )r 2 — (1 — a) = 0 .
Since h(Q)Ji(l) < 0, then this equation has for 0 < a < 1 one and only one root r in the interval [0,1], given by (3.1).
If r < r*, then the equation h(r) = 0 has foi* | < a < 1 the root r in the interval [0,1], given by (3.2).
Since the function r*(a) given by (3.12) increases in interval [0,1]
and the function r(a ) given by (3.2) decreases in interval [|, 1], because of r*(|)< ?*(£) and r * ( l ) > r ( l ) there exists a number a0, | < a0 < 1 such that r(a) < r*(a) for a0 < a < 1. Eliminating r from (3.12) and (3.2) we obtain a0 given by (3.3). Since r (a) given by (3.1) decreases for 0 < a < 1 and r (0) > r*(0), r (1) < r*(l), then because of f (a0) = r*(a0) we obtain r (a) > r*{a) for 0 < a < a 0. Hence, in view of g{r) > h(r) we conclude that i*. с. V (a), the radius of convexity of У (a), satisfies the condition
r. c. V ( cl ) ^ r 0 with
I r for 0 < a < a0, r for a0 < a < 1.
2) We shall prove that
г. с. У {a) < r 0.
We observe first that if a function P* {z) of p 2 («) for 0 < a < a0 satisfies condition (3.9) at some point z* = r e t(p with s = $x(r) and t = 0, then
(3.16) P * (r-e i<p) = s 1(r),
where s^ r) is given by (3.11).
Denote by A* and the values of the parameters A and ek (Jc = 1, 2) corresponding to the function P*(z) (comp. (2.5)-(2.6)). Because of assumption that rer)-e~ü = 1 and since t = 0, we obtain rj = 1. Thus, from (2.18), in view of (2.17) and (2.10)
(3.16) 1 — e* • e* e2i(p = ег>(£*-e*x)rA*.
Simultaneously, because of
(3.17) im P* (r • ei<p) = 0,
we have from (2.8) im у — 0. Thus by (2.9) and because of r\ = 1 we obtain
(3.18) f (i - £; £* е2г>) - é* (4 - £^) /*.
Badius of convexity
381
By (3.16) and (3.18)
(3.19) ( 1 - f 2) (£*-£*) A* = 0.
From (3.19) we get A* = 0 or £* = £*. Supposing that the second of these equalities holds, we would have because of (2.10) that x = 2fi*, where e* = 4 for h = 1, 2, hence by (2.9) Tc = q ( t ) and y = £*ег<р. Thus (3.20) P * (r-e i(p) — l + ars*eiq>.
By (3.17) and (3.20) we would have e*ег<р = ± 1 and because of Si(r) < 1 we obtain that в*егч> = — 1. Therefore
(3.21) P *(r -ei4>) = 1 - a r . From (3.16) and (3.21) we get
( 2 a - l ) r 2- 2 f + 1 = 0,
which is true if and only if a = a 0. Hence A* = 0 and because of (3.16) Thus
where
Gi Sn e- 2i<P'
P*(Z) (l + a)ff*(z) + l — a p*{z) + l
P*(*) _ i
1 — £* Z
1 -f £2 ^ \ i - 4 ч ‘ Finally we obtain
(3.22) P*(z)
where
1 — (1 — a)te l<pz — ae 2t,pz2 1 - t e ' i(pz
(3.23) t = ±(e*ei,p + e*e-i<p).
We shall find t. By (3.1) and (3.11) we obtain
^ a _______ __
s\(r) = ---(l + a+]/l + 2a —3a2), 2
then
(3.24) P* (r-ei<p) = J [ l - a + l / l + 2 a -3 0 * ].
On the other hand, from (3.1)
a — 1 - f V l - f 2a — 3a2 (3.25)
2a
382 J. K a c z m a rs k i
Substituting У1 + 2а —За2 from (3.25) into (3.24) we obtain
(3.26) P * (r-e i(p) = —a.
From (3.22) for z — r -et<p we have
(3.27) P * (r-e i,p) 1 —(1 — a)t-r — a r 2 1 — t-r
Thus, by equating P*{rei(p) in (3.26) and in (3.27) we obtain t given by (3.6).
Denote b j f*(z) a function of the form г + a2 2 2 + ... regular in К and such that
/ * » = P *(z),
Z e K .The function P*{z) has the pole zx = —-ег,р with zx4 К and f*'(z) t
is a regular function in the disc K , thus the integrals of these functions exist along any regular curve Г с К with the origin and with the end
point at 0 and z, respectively, where ze K.
Thus we conclude from (3.22) that
„ , % /•' 1 - (1 - a) to"" f - ae~2,> £* ^
i ( z ) - J --- ---ЙС’ 0
i. e. f*{z) is of the form (3.4), where f 0(z) is given by (3.5). Evidently, f* (z )e V (a ).
Differentiating the function P*(z) we obtain after some calculation zP*' (z) te~birpzb - 2 e~2i<pz2 + te~i<pz
P*(z) = “ (1 - te~i<pz) [1 - (1 - a) te~i<pz - ae~2i,pz2] * Thus
re 1
Г (г-е * * ) = 1 + a t-r3 — 2 -r2 + t-r ( l — t ‘r ) [ l — ( l — a)t'T — ar2]
Since
f .p — 2 -r2 (r2 + l) —2 -f2 )
1 - t - r 1 —r2 and
1 — (1 — a )t-r — ar2 = — ar4, + (2a — l ) f 2 + 1 — a
= a,
Radius of convexity
