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40Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 14, 2008
UDK 517
ON PROBLEMS OF UNIVALENCE FOR THE CLASS TR(1/2)
M. SOBCZAK-KNEC, P. ZAPRAWA
In this paper we discuss the class TR(2) consisted of typically real functions given by the integral formula
f w = /, d,‘(,)'
where ^ is the probability measure on [— 1,1]. The problems of local univalence, univalence, convexity in the direction of real and imaginary axes are examined. This paper is the continuation of research on TR(|), especially concerning problems, which results were published in [5].
Let A denote the set of all functions which are analytic in the unit disk A = {z G C : |z| < 1} and normalized by f (0) = f '(0) — 1 = 0. Let TR denote the well known class which consists of typically real functions. Recall that the function f G A belongs to TR if and only if the condition
9z • 9f (z) > 0 z G A.
is satisfied.
Rogosinski [4] proved that f G TR f (z) = kt(z) d^(t),
where kt(z) = 1_2Zt+z2, and ^ belongs to P[-i,i], i.e. the collection of all probability measures on [—1,1]. Similarly Szynal [6] defined the class
TR(2) = {f gA : f (z) = f-1 ft(z) d^(t), ^ G P[-1,1^ , where
f-'z) = z ()2 = TT-llrfz* ■ <1»
© M. Sobczak-Knec, P. Zaprawa, 2008
In this paper Szynal considered the coefficients problems. He proved that if f (z) = z + ^=2 anzn is in TR( 1) then |an| < 1. This fact means
that the coefficients of the function f G TR( 2) are bounded by the same number as the coefficients of functions in the classes CV, CV(i), ST(2) consisting of convex functions, convex in the direction of the imaginary axis functions, and starlike of order 1 functions, respectively. Moreover, he proved that the functions of the class TR( 1) are typically real, so TR( 1) C TR.
We shall point out the property, which in essential manner differs the class TR(2) from the class TR. However the functions kt of the class TR are starlike, the functions of the form
aki + (1 — a)k-i , a G (0, 1)
are not univalent. These functions are extremal in many univalence problems. One of the most important functions is the function
z(1 + z2)
z ^ [ki(z) + k-i(z)]/2 = (( —+z2)2) ,
which is used, for example, to determining the domain of univalence or the domain of local univalence for TR.
By analogy to the class TR, the kernel functions ft of the class TR( 2) are starlike of order 1/2. On the other hand, it is easy to check, that the functions given by the formula
afi + (1 — a)f-i , a G [0,1]
are univalent and convex in the direction of the imaginary axis. Hence, these functions are not extremal in the problems concerning univalence.
The classes CVR, CVR(i), STR(2) (where AR denotes the subclass
of a class A consisting of functions having real coefficients) and the class TR(i) are connected by the following inclusions, namely
CVR C STR ^0 C TR ^0 (2)
and
CVR C CVR(i) C TR(1). (3)
The relations (2) result from the equality coSTR (!) = TR (2) given by Hallenbeck [1], where co A denotes the closed convex hull of A, and the well known theorem of Marx and Strohhacker .
The fact coCVR = CVR(i) (compare [5]), the relation (2) and convexity of the class TR (2) (see [6]) give us (3).
Now, we are going to prove that the class TR( i) is the essential superclass of CVR, CVR(i) and STR( i). In order to do this we shall find functions belonging to TR(2) which are not univalent.
Let us consider the functions
Ft(z) = [ft(z) + f-t(z)] /2, t G [0,1] .
Theorem 1. For all t g (0,1) there exist rt g (0,1) such that functions Ft are not locally univalent in Ar, r > rt.
Proof. Let t g (0,1). We have F/(z) = 2
i-tz i i+tz
3 I 3
(i-2tz+z2) 2 (i+2tz+z2) 2
Hence, the equality
F/(ir) = 0 is equivalent to
K(1 — itr)(1 — r2 + 2tir)2 = 0. (4)
Using
\J1 — r2 + 2tir = \j — ^1 — r2 + J(1 — r2)2 + 4t2r2'j +
+i\j^ (—1 + r2 + \/(1 — r2)2 + 4t2r2j (5)
the condition (4) could be written as
[1 — r2 — tr(1 — 2tr + r2)] [1 — r2 + tr(1 + 2tr + r2)] +
(1 — r2)(1 + t2r2)/(1 — r2 )2 + 4t2r2 = 0.
Let us denote the left hand side of (5) by G(t,r). The function G is continuous with respect to both variables. Moreover, G(t, 0) = 2 and G(t, 1) = —4t2 (1 — t2 ) < 0 for t G (0, 1). We conclude that there exist
rt G (0,1) such that G(t,rt) = 0.
Now, we determine the smallest number rt, which was described above. Solving the system of equations
J G(t, r) = 0 I ft (t, r) = 0
we obtain J
G(t, r) = 0
t2 = -5+6r2+3r4 1 ~ 8r2 .
Hence (1 + r2)3(7 — 9r2) = 0 and consequently r = ^ =0, 88.... We have proved that
Corolary. The radius of locally univalence rLU of TR( 2) satisfies the condition rLU < ^.
This means that there are the functions of the class TR( 2) which are not univalent in each disk Ar, r > ^.
In the proof of the following theorems we will apply the Krein-Milman Theorem. This theorem concerns the extremalization of linear and continuous functionals in a given A C A. By this theorem, such real functionals attain the lowest and the greatest values on the extreme points of A.
Theorem 2. If f g TR (2) then K> 2 for z g A.
In the proof of Theorem 2 we use the following lemma.
Lemma 1. Let = (^j2. Then f g ST ^ g g ST(2).
Proof of Theorem 2.: The functional 3?f(z) is linear and continuous
z
so
min | K ,f G TR ^ = min | Kft(z) ,t G [—1,1]
Let ft be given by (1). From Lemma 1 it follows that there exists the function gt G ST(i) which satisfies = \Jgtzz). Using the known
inequality Ky^-^ > i for h G ST we obtain the conclusion of this theorem.
Theorem 3. The radius of bounded rotation rP/ of TR(i) is equal to rP/ = = 0, 707...
Proof. From the Krein-Milman Theorem we have
1
min |Kf'(z), f G TR (^ , |z| = r j > 0 <
min (Kft/(z),t G [—1,1], |z| = r} > 0.
Let ft be given by (1). Since ft G ST (i), there is
zft'(z^ 1
ft(z)
1z
It means that there exists a function wi of the class B = (w G A : w(0) = 0, |w(z)| < 1,z G A} such that
zft (z) 1
ft(z) 1 — wi(z)
Hence, we have
f/(z) =
1
ft(z)________________
z 1 — wi(z)
(6)
From Theorem 1 it follows that
and consequently
K > 1, z G A, z2
ft(z) , 1
z 1 — z
Therefore, there exists a function w2 G B such that
ft(z)
1
z 1 — W2(z)
Finally, the function ft/ can be written in the form
ft'(z)
1
1
1 — W2(z) 1 — wi(z)'
(7)
(8)
The condition Kf/(z) > 0 is equivalent to the condition |Arg f/(z)| < ^2. Using (8) and simple estimation we have
|Arg ft(z)
Arg
1
1
1 — W2(z) 1 — wi(z)
< max 2
Arg
1
1 — w(z)
< 2 arcsin | z|
Hence, if 2arcsin |z| < n or equivalently |z| < sin n then Kf(z) > 0.
The equality in (??) appears for wi(z) = z, w2(z) = z. Hence, from (7) we get the function ft(z) = for which Kf/(z) has negative values while
|z| > #. Z
Theorem 4. The radius of convexity in the direction of the imaginary axis rCV(j) of TR(2) is equal to rCV(j) = \J2^3 — 3 = 0, 68...
Proof. It is known that, if f g A then
f G CVR(i) ^ zf/(z) G TR.
Hence
f G CVR(i) ^ SzSzf/(z) > 0, z G A.
Let z G A, Sz > 0 and f G TR (2).
From the Krein-Milman Theorem
min |Szf/(z), f G TR ^^ , z G A j = min (3zft'(z), t G [—1,1], z G A} ,
where ft is given by (1).
Now we use the theorem established by MacGregor in [3]
Theorem A. If f G ST(2) then f (Ar) is convex for r < /2^3 — 3. Since ft G ST (2), from Theorem A in particular it follows that the set ft(Ar) is convex in the direction of the imaginary axis for r < \J2^3 — 3.
We are going to prove that for r > \J2^3 — 3 there exists a function ft0 of the form (1) such that SzSzf/ (z) < 0 for some z G Ar.
Let Gt(z) = zft/(z). We have Gt(reiV) = reiV----------------i-treiy--------3. The
^ J tK ' (i_2trei^+r2e2i^) I
argument of the tangent vector to the curve r = dGt(Ar) in the point Gt(r) is equal to
i dGt , , \ , , ,, n . ,
arg I (r) I = arg (* • wt(r)) = ^ + arg wt(r),
i / \ r(i-tr-2r2 + t2r2+tr3)
where wt(r) = —-------------------!-r- .
(i-2tr + r2) 2
The inequality wt(r) > 0 is true for all t G [—1,1] if r < \J2^3 — 3. For r > V2^3 — 3 and to = i-r the inequality wt0 (r) < 0 holds.
It means that for r > \J2^3 — 3 the argument of the tangent vector to r in Gto (r) is equal to —n. Hence, there exists <^0 such that
3Gt0 < 0 for ^ G [0, ^o) .
Furthermore, f (Ar) is convex in the direction of the imaginary axis in the disk |z| < \J2^3 — 3 and this number is best possible. The extremal function is z
fto (z) = I 1 z2 = .
^1 — 2 z + z2
Using the similar method to that from the proof of Theorem 2, we estimate the radius of convexity in the direction of the real axis in TR(i). Koczan in [2] determined the representation formula for the class CVR(1). Namely
Theorem B. The function f belongs to CVR(1) if and only if f G A, f is real on ( — 1, 1), and there exists p G [0, n] such that
K [(1 — 2zcosp + z2)f/(z)] > 0, z G A.
We make use of the following fact
m^lArg 14 :KI<W< 11 = 2arcsin lrl' (9)
Indeed, from the maximum principle for analytic functions we have
max j^Arg1-—z : |Z| < |z| < ^ = max jArg!—^ : |z| = |Z| < ^ .
Using twice the inequality Arg(1 — w) < arcsin |w| for w G A we obtain (9) .
Theorem 5. The radius of convexity in the direction of the real axis rCV (i) of TR( 2) satisfies the inequality sin 8 =0.38 ... < rCV (i) < %/2—1.
Proof. Let f g TR(2).
Then 2
K [(1 — 2z cosp + z2)f/(z)] =J K [(1 — 2zcosp + z2)ft/(z)] d^(t).
From (8) we have
2. . 1 — 2z cos P + z2
(1 — 2zcosP + z )ft(z) =
(1 — wi(z))(1 — W2(z))’
where wi, w2 G B.
Let us consider the inequality
. 1 — 2z cos P + z2
Arg -
(1 — wi(z))(1 — W2(z)) or equivalently
4 1 — ze-i^ 1 — ze^
Arg
1 — wi(z) 1 — W2(z)
n <2,
< n. (10)
2
We have |wk(z)| < |z|, k = 1, 2. From (9) it follows now that if 4arcsin |z| < n then the inequality (10) is satisfied. Consequently, if |z| < sin 8 then
K [(1 — 2zcosP + z2)ft/(z)] > 0 . (11)
This and Theorem B leads to rCV(i) > sin 8. The extremal function in
the inequality (11) does not have real coefficients so
n
rev(i) > sin ^.
Moreover, for the function
fW = T—z2 = 1 (r+T + 1—z) G TR(i) (12)
the set f (Aro), ro = a/2 — 1 is convex in the direction of the real axis and
the number r0 is best possible. It results from the fact that the function
f (iz) z
1 + z2
is convex in the direction of the imaginary axis in the set i • H = |rej0 : 1 — r2 > 2r| cos 0|}. Hence, the function (12) is convex in the direction of the real axis in the set H of the form |rej0 : 1 — r2 > 2r| sin 0|}. Therefore rCVR(i) < V2 — 1.
From given above theorems we obtain the corollaries concerning star-likeness and convexity of functions from TR( 1).
Corolary. The radius of starlikeness rST of TR( 1) satisfies the inequality < г«т < it •
The left hand side inequality results from the fact that the functions of the class {/ Є A : Ж> 1, z Є A} are starlike in the disk A^2, (see [7]) and from Theorem 2. The upper estimation is the consequence of the inequality proved in Theorem l.
From Theorem 4 we obtain
Corolary. The radius of convexity rev of TR( 2) satisfies the inequality rev < У2 - l.
Bibliography
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[3] McGregor M. T. On three classes of univalent functions with real coefficients / M. T. McGregor // J. London Math. Soc. 39. (1964). 43-50.
[4] Rogosinski W. W. Uber positive harmonische Entwicklungen und typisch-reele Potenzreihen / W. W. Rogosinski // Math. Z. 35. (1932). 93-121.
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Department of Applied Mathematics Lublin University of Technology
ul. Nadbystrzycka 38D 20-618 Lublin, Poland
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