# Journal of Inequalities in Pure and Applied Mathematics

advertisement ```Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 4, Article 79, 2003
PARTIAL SUMS OF FUNCTIONS OF BOUNDED TURNING
JAY M. JAHANGIRI AND K. FARAHMAND
K ENT S TATE U NIVERSITY B URTON ,
O HIO 44021-9500, USA.
[email protected]
U NIVERSITY OF U LSTER ,
J ORDANSTOWN , BT37 0QB,
U NITED K INGDOM
[email protected]
Received 15 July, 2002; accepted 10 June, 2003
Communicated by N.E. Cho
A BSTRACT. We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.
Key words and phrases: Partial Sums, Bounded Turning, Libera Integral Operator.
2000 Mathematics Subject Classification. Primary 30C45; Secondary 26D05.
1. I NTRODUCTION
Let A denote the family of functions f which are analytic in the open unit disk U = {z :
|z| &lt; 1} and are normalized by
(1.1)
f (z) = z +
∞
X
ak z k , z ∈ U.
k=2
For 0 ≤ α &lt; 1, let B(α) denote the class of functions f of the form (1.1) so that &lt;(f 0 ) &gt; α
in U. The functions in B(α) are called functions of bounded turning (c.f. [3, Vol. II]). By the
Nashiro-Warschowski Theorem (see e.g. [3, Vol. I]) the functions in B(α) are univalent and
also close-to-convex in U.
For f of the form (1.1), the Libera integral operator F is given by
Z
∞
X
2 z
2
F (z) =
f (ζ)dζ = z +
ak z k .
z 0
k
+
1
k=2
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
080-02
2
JAY M. JAHANGIRI AND K. FARAHMAND
The n-th partial sums Fn (z) of the Libera integral operator F (z) are given by
Fn (z) = z +
n
X
k=2
2
ak z k .
k+1
In  it was shown that if f ∈ A is starlike of order α, α = 0.294..., then so is the Libera
integral operator F. We also know that (see e.g. ), there are functions which are univalent
or spiral-like in U so that their Libera integral operators are not univalent or spiral-like in U.
Li and Owa  proved that if f ∈ A is univalent in U, then Fn (z) is starlike in |z| &lt; 83 . The
number 38 is sharp. In this paper we make use of a result of Gasper  to provide a simple proof
for the following theorem.
Theorem 1.1 (Main Theorem). If 14 ≤ α &lt; 1 and f ∈ B(α), then Fn ∈ B 4α−1
.
3
2. P RELIMINARY L EMMAS
To prove our Main Theorem, we shall need the following three lemmas. The first lemma is
due to Gasper ([2, Theorem 1]) and the third lemma is a well-known and celebrated result (c.f.
[3, Vol. I]) which can be derived from Herglotz’s representation for positive real part functions.
Lemma 2.1. Let θ be a real number and m and k be natural numbers. Then
m
1 X cos(kθ)
(2.1)
+
≥ 0.
3 k=1 k + 2
Lemma 2.2. For z ∈ U we have
m
X
zk
&lt;
k+2
k=1
!
1
&gt;− .
3
Proof. For 0 ≤ r &lt; 1 and for 0 ≤ |θ| ≤ π write z = reiθ = r(cos(θ)+i sin(θ)). By DeMoivre’s
law and the minimum principle for harmonic functions, we have
!
m
m
m
k
X
X
z
rk cos(kθ) X cos(kθ)
=
&gt;
.
(2.2)
&lt;
k+2
k+2
k+2
k=1
k=1
k=1
Now by Abel’s lemma (c.f. Titchmarsh ) and condition (2.1) of Lemma 2.1 we conclude that
the right hand side of (2.2) is greater than or equal to −1
.
3
Lemma 2.3. Let P (z) be analytic in U, P (0) = 1, and &lt;(P (z)) &gt; 12 in U. For functions Q
analytic in U the convolution function P ∗ Q takes values in the convex hull of the image on U
under Q.
operator “∗” stands
for the Hadamard product or convolution
power series f (z) =
PThe
P∞
P∞ of two
∞
k
k
k
a
z
and
g(z)
=
b
z
denoted
by
(f
∗
g)(z)
=
a
b
z
.
k=1 k
k=1 k
k=1 k k
3. P ROOF OF THE M AIN T HEOREM
≤ α &lt; 1. Since &lt;(f 0 (z)) &gt; α we have
!
∞
X
1
1
&lt; 1+
kak z k−1 &gt; .
2(1 − α) k=2
2
Let f be of the form (1.1) and belong to B(α) for
(3.1)
J. Inequal. Pure and Appl. Math., 4(4) Art. 79, 2003
1
4
http://jipam.vu.edu.au/
PARTIAL S UMS OF F UNCTIONS OF B OUNDED T URNING
3
Applying the convolution properties of power series to Fn0 (z) we may write
n
X
2k
0
Fn (z) = 1 +
(3.2)
ak z k−1
k
+
1
k=2
!
!
∞
n
X
X
4
1
kak z k−1 ∗ 1 + (1 − α)
= 1+
z k−1
2(1 − α) k=2
k
+
1
k=2
= P (z) ∗ Q(z).
From Lemma 2.2 for m = n − 1 we obtain
!
n
X
z k−1
1
(3.3)
&lt;
&gt;− .
k+1
3
k=2
Applying a simple algebra to the above inequality (3.3) and Q(z) in (3.2) yields
!
n
X
4 k−1
4α − 1
&lt;(Q(z)) = &lt; 1 + (1 − α)
z
&gt;
.
k
+
1
3
k=2
On the other hand, the power series P (z) in (3.2) in conjunction with the condition (3.1) yields
&lt;(P (z)) &gt; 12 . Therefore, by Lemma 2.3, &lt;(Fn0 (z)) &gt; 4α−1
. This concludes the Main Theorem.
3
Remark 3.1. The Main Theorem also holds for α &lt; 41 . We also note that B(α) for α &lt; 0 is no
longer a bounded turning family.
R EFERENCES
 D.M. CAMPBELL AND V. SINGH, Valence properties of the solution of a differential equation,
Pacific J. Math., 84 (1979), 29–33.
 G. GASPER, Nonnegative sums of cosines, ultraspherical and Jacobi polynomials, J. Math. Anal.
Appl., 26 (1969), 60–68.
 A.W. GOODMAN, Univalent Functions, Vols. I &amp; II, Mariner Pub. Co., Tampa, FL., 1983.
 J.L. LI AND S. OWA, On partial sums of the Libera integral operator, J. Math. Anal. Appl., 213
(1997), 444–454.
 P.T. MOCANU, M.O. READE AND D. RIPEANU, The order of starlikeness of a Libera integral
operator, Mathematica (Cluj), 19 (1977), 67–73.
 E.C. TITCHMARSH, The Theory of Functions, 2nd Ed., Oxford University Press, 1976.
J. Inequal. Pure and Appl. Math., 4(4) Art. 79, 2003
http://jipam.vu.edu.au/
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