CC BY
89Probl. Anal. Issues Anal. Vol. 7 (25), No. 2, 2018, pp. 131-143 131
DOI: 10.15393/j3.art.2018.5830
UDC 517.54
A. Sairam Kaliraj
INJECTIVITY OF SECTIONS OF CLOSE-TO-CONVEX HARMONIC MAPPINGS WITH FUNCTIONS CONVEX IN ONE DIRECTION AS ANALYTIC PART
Abstract. In this article, we prove a two-points distortion theorem and obtain sharp coefficient estimates for the families of close-to-convex harmonic mappings whose analytic part is a function convex in one direction. By making use of these results, we determine the radius of univalence of sections of these families in terms of zeros of a certain equation. the lower bound for the radius of univalence has been obtained explicitly for the case a = 1/2. Comparison of radius of univalence of the sections has been shown by providing a table of numerical estimates for the special choices of a.
Key words: Univalent Harmonic, convex in one direction, close-to-convex, partial sums, sections
2010 Mathematical Subject Classification: Primary: 30C45; Secondary: 31A05, 30C55
1. Introduction and motivation. Harmonic mappings are useful in the study of fluid flow problems (see [3]). Furthermore, univalent harmonic functions having special geometric properties such as convexity, starlikeness, and close-to-convexity arise naturally while dealing with planar fluid dynamics problems. For example, in [3, Theorem 4.5], Aleman et al. considered a fluid flow problem on a convex domain satisfying an interesting geometric property. Furthermore, harmonic mappings which appear as solutions to some of these real-world problems are very complicated and, hence, evaluating values of such functions is challenging. In this connection, it is interesting to consider the problem of approximating harmonic mappings by harmonic polynomials without compromising the univalency of the polynomial. With this brief motivation, we shall begin to consider the partial sums for the univalent analytic functions and univalent harmonic mappings.
© Petrozavodsk State University, 2018
Let S denote the class of all normalized univalent analytic functions 0 defined in the unit disk D = {z : |z| < 1} such that
0(z ) = z + ^ ak zk. (1)
k=2
-.th
For n ^ 2, the n partial sum of 0 is the polynomial defined by
n
sra(0)(z) = z + ^ ak zk.
k=2
In [23] Szego proved that the partial sum sn(0) is univalent in |z| < 1/4 for all 0 6 S and n ^ 2. In [19], Robertson proved that the nth partial sum of the Koebe function is starlike in the disk |z| < 1 — 3n-1 logn, for n ^ 5, and that 3 cannot be replaced by a smaller constant. The general theorems on convolutions [20] (see also [8, p. 256, p. 273]) allow to infer that sn(0) is convex, starlike, or close-to-convex in the disk |z| < < 1 — 3n-1 log n, for n ^ 5, whenever 0 is convex, starlike, or close-to-convex in D (for details on the class S and its geometric subclasses, one can refer [8]), respectively. However, the exact radius of univalence rn of sn(0) remains an open problem, if 0 6 S. Jenkins [11] proved that sn(0) is univalent in |z| < rn for 0 6 S, where the radius of univalence rn is at least 1 — (4 log n — log(4log n))/n for n ^ 8. However, by making use of the exact coefficient bounds, one could get rn> 1 — (4logn — 2log(logn))/n for n ^7. Moreover, 1 —(4logn—2log(logn))/n > 1 —(4logn—log(4logn))/n for n ^ 55.
Denote by H the class of all complex-valued harmonic functions f = h + g in D normalized by h(0) = g(0) = 0 = h'(0) — 1. We call h and g the analytic and the co-analytic parts of f, respectively, and, obviously, they have the following power series representation:
h(z) = z + akzk and g(z) = ^ bkzk, z 6 D.
k
k=2 k=1
We shall use this representation throughout the discussion. The Jacobian Jf of f = h + g is Jf(z) = |h'(z)|2 — |g'(z)|2. We say that f is sense-preserving in D if Jf (z) > 0 in D. Let SH denote the class of all sense-preserving harmonic univalent mappings f 6 H and set SH = {f 6 SH : fz(0) = 0}. For many basic results on univalent harmonic mappings in SH
*0 H ,
and CH that
and its well-known geometric subclasses, namely, K°H, S map D onto convex, starlike, and close-to-convex domains, respectively, we refer to [7,15]. Here we recall a new subclass of SH, namely, SH (S) introduced in [16], where
SH(S) = {h + g e SH : h + eieg e S for some B e R} .
This class along with a related conjecture concerning convex harmonic mappings were proposed in [16]; this conjecture was later disproved by Kayumov et al in [12]. For f = h + g e SH with the power series representation as in (2), the sections/partial sums sn,m(f) of f are defined as
sn, m (f)(z) = sn (h)(z) + sm(g)(z),
where n ^ 1 and m ^ 2. However, the special case m = n ^ 2 seems interesting in its own merit. From the definition above it is clear that partial sums of f can be thought of as an approximation of f by complex-valued harmonic polynomials and, thus, approximation of univalent harmonic mappings by univalent harmonic polynomials might lead to new applications. For fundamental results on the partial sums of univalent harmonic mappings, one can refer to [13,14,17,18]. The coefficient problem for SH remains open. For large values of n, coefficient estimates are given in [18]. On the other hand, in a recent paper [1] Ali et al. have shown that if f = h + g e SH, then |a2| ^ 20.92 which is the best known estimation. We recall a few results from [17] which are motivation for the problem which we consider in this article.
Theorem A. Let f = h + g e SH with the series representation as in (2). Suppose that f belongs to C°H, the class of close-to-convex harmonic mappings or SH(S). Then the section sn,m(f) is univalent in the disk |z| < rn,m. Here rn,m is the unique positive root of the equation ip(n,m,r) = 0, where
^(n,m,r) =
1 — r 1 + r
1
with
where
Rn ^ ^ Akr , Tm
k=n+l
1—r 1 + r
£ A
k=m+l
- R -T
nm
kr
k1
Ak
k(k + 1)(2k + 1) 6 '
3
6
In particular, every section sn,n(/)(z) is univalent in the disk |z| < rn,n, where
r„„ >r„\ := 1 - (7log n - 4log(log n)) for n > 15.
' n
Moreover, rra>TO ^ rLz, where l = min{n,m} ^ 15.
We have the following interesting result for functions in the convex family KH of harmonic mappings: the lower bound of rra>TO can be improved compared to that of the bounds in Theorem A.
Theorem B. Let / = h + g G KH with the series representation as in (2). Then the section sn,m(/) is univalent in the disk |z| < rn,m, where rn,m is the unique positive root of the equation ^(n,m,r) = 0. Here
^(n, m, r)
1r
(1 + r)3 ^
k(k +1) rfc-l"
2
E
fc=m+1
k(k - 1) rk-i
(3)
In particular, for n ^ 5, and 9 G R, the harmonic function
sn,n(/; 9)(z) = sra(h)(z) + sra(g)(z)
is univalent and close-to-convex in the disk |z| < 1 — 3n-1 logn. Moreover, we have rn,m ^ 1 — (4logl — 2log(logl))/l, where l = min{n,m} ^ 7.
For —1/2 ^ a < 1, set F (a) = j h + g G H : Re ^1 + 1 > a,
g^ = z, V z G D h' (z)
and F = F(—1/2). Note that for —1/2 ^ a < 1, / G F(a) has the following integral representation (see [2]):
/(z)
1
0 |x| = 1
(1 — xt)2(1-a)
d^(x)dt +
0 |x| = 1
(1 — xi)2(1-«)
—
for some probability measure ^ defined on dD := {z G C : |z| = 1}.
Remark 1. Let us consider the family of close-to-convex harmonic functions F. A keen observation of the proof of Theorem B reveals that the results in Theorem B are valid for the class F too (note that members
z
z
e
of F are not necessarily even starlike). Comparison of this fact with Theorem A motivates us to consider certain subclasses of close-to-convex harmonic mappings with special condition on the analytic part of f.
In this article, we determine the radius of univalence rn,m of sn,m(f), when f e F(a), where a e [-1/2,1). The lower bound for rn,m is determined for the special choice of a. The paper is organized as follows. In Section 2, we recall a few known results from the literature, which are helpful in the proof of our main theorems. We present our main theorems in Section 3.
2. Useful results. The following result by Bazilevich [5] gives the necessary and sufficient condition for a normalized analytic function to be univalent in D.
Theorem C. An analytic function < defined in D and determined by (1) is univalent in D if and only if for each z e D and each t G [0,n/2],
<(rein) - <(re^) rein_
E
k= 1
ak-
sin kt sin t
kk-1
= 0,
(4)
where t = (rq — ^)/2, z = rei(v+^/2 and
sin kt
sin t
k.
t=0
In [21], Starkov generalized this result to the class of normalized sense-preserving harmonic mappings in the following form.
Theorem D. A sense-preserving harmonic function f = h + g defined in D determined by (2) is univalent in D if and only if for each z e D \ {0} and each t e (0,n/2],
f (rein) — f (re^)
rein_rei^
E
k=1
(ak zk — bk zk)
sin kt sin t
= 0,
(5)
where t = (rq — ^)/2 and z = rei(v+^/2.
Further extension of Theorem D along with its use may be obtained from the article of Amozova et al. [4]. The following two-points distortion theorem plays a crucial role in the proof of our main results.
Theorem E. If f eS, r e (0,1), G R such that t = ^, then
f (reit) — f (re^)
re
it
' rei^
>
1- r2
If (reit )|f (re^ )|.
3. The distortion theorem and the partial sums problem. By
making use of Theorem E, we derive a two-points distortion theorem for functions in the family F(a). This result is crucial in the proof of our main theorem.
Theorem 1. (Two points distortion theorem) Suppose that f = h + g G F(a) for some a G [-1/2,1). For each A G C such that |A| = 1, define
Fa(z) := h(z) + Ag(z).
Then for 0 < r < 1 and t, — G R such that t = —, f satisfies the following inequalities
FA(reii) - FA(re^)
reit_re^
f (rei4) - f (re^)
1 - r2
L2(r, a) =: A(r, a),
(6)
where
re
2r
it_re#
^ A(r, a) for 0 < r < 1,
- log(1 + r)
L(r, a) := <
1 + r
2 log(1 + r) - r
(1 + r + 2a(1 - r))(1 + r)2a-1 - (1 + 2a)
if a if a
0,
1/2,
2a(2a - 1)
if a G [-1/2,1) \{0,1/2}.
Proof. Let f = h + g G F(a) for some a G [-1/2,1) and FA(z) := h(z) + Ag(z). Set fA = h + Ag. It is clear that both FA and fA are univalent and close-to-convex in D for all A such that |A| = 1, which follow from a result of Bharanedhar et al. [6]. For every pair of points z1 = rei4 and z2 = rei^, we can find a A such that
(h(zi) - h(z2)) + (g(zi) - g(z2)) = (h(zi) - h(z2)) + A(g(zi) - g(z2)). Therefore, by using Theorem E, we get
f (rei4) - f (rei^) F\(reit) - F\(re^)
re
it _ . r ei^
re it_rei^
1 - r2
|Fa (reit) | ■ |F\(rei^ )|.
In order to complete the proof, we need to find the lower bounds of F(z)|.
Let y be the preimage under Fx of the line segment joining 0 and Fx (z). Then
F(z)| = / F(Z)Hd(| >J(|h'(c)I-Ig(Z)l)-|dCI ^
Y Y
r
> I(! -IZI)"Ih'(Z)HdZI (1 +1 dp> (8)
Y 0
where the last inequality is a consequence of the following well known distortion theorem for the convex function of order a (see [9, p. 139, Vol. 1, Theorem 1] for the case a such that 0 ^ a < 1))
Ih'(z)I ^ (1 + ^(l-a) ' r = |Z| < 1.
Also, for a G [-1/2, 0), the above inequality holds true, which can be obtained using the same proof technique. The desired conclusion follows, if we use the inequality (8) in (7). □
Next, we provide the sharp coefficient estimates for the functions in the family F (a).
Theorem 2. Suppose that f = h + g G F (a) for some a G [-1/2,1) with the series representation as in (2). Then, for all n ^ 2, the coefficients of f satisfy the following inequalities:
n1
IanI ^ An(a) and IbnI ^ -— An(a), (9)
n 2a
where
n
An(a) = -U(j - 2a). (10)
AH these bounds are sharp and each inequality becomes equality for the close-to-convex functions fa(z) = ha(z) + ga(z) and their rotations, where
fa(z) = / (1 - t)2(1-a) dt + ^ /' (1 - t)2(1-a) dt (11)
00
z
Proof. The proof follows from the coefficient bounds for the convex functions of order a (see [9, p. 140, Vol. 1, Theorem 2]) and a result of T. J. Suffridge [22, Theorem 5.6]. We remark that the original proofs given in the above references are considered for the values of a G [0,1), which can be extended to the case of a G [-1/2, 0); we skip the details here as the computations are similar (See also [24, Lemma 3.1, Theorem 3.2]). □
Theorem 3. Let f = h + g G F (a), for some a G [-1/2,1) with the series representation as in (2). Then, for 9 G R, the harmonic function
Sn,m(f; 9)(z) = sra(h)(z) + e' sm(g)(z)
is univalent in the disk |z| < rn,m, where rn,m is the unique positive root of the equation ^(n, m, r, a) = 0 in (0,1). Here
<x
„fc-1
k(k 1) Afc(a)rk-1
(k - 2a)
^(n,m,r,a) = A(r,a)— ^ [fcAfc(a)rk - ^
fc=n+1 fc=m+1
(12)
where A(r, a) and Ak(a) are defined as in (6) and (10), respectively. When a = 1/2, we have rn,m ^ 1 — (2log/)//, where I = min{n,m} ^ 3.
Proof. Suppose that f = h + g G F(a) for some a G [—1/2,1) with the series representation as in (2). For 0 < r < 1 and 9 G R, we set
(z) = h(rzl + gM
z z
and one could see that
oo
Fr,0(z) = z + afcrk-1zk + e'^Y^ rk-1zk. fc=2 fc=2
It is clear, from the definition of Fr,g, that finding the largest radius of univalence of sn,m(f; 9) for every value of 9 G [0, 2n] is equivalent to finding the largest value r such that sn,m(Fr,6>) is univalent in D for every 9 G [0, 2n] (See [10] for the details). Using Theorem C, we see that sn,m(Fr,6>) is univalent in D if and only if the associated section Pn,m,r,a(z) has the property that
k=1
M r 'hi
II k , 7 / k\ Sin kt
(a'k zk + bk zk )—— v k k sin t
= 0, for all z G D \ {0},
and t G [0,n/2], where M = max{n,m}, l = min{n,m}, a'k b'k = bk rk-1 for all k ^ l,
«fc r
k1
ak =
a,k r 0
k-1 for all k > l if M = n, for all k > l if M > n,
bk
eid bk rk-1 for all k>l if M = m, 0 for all k > l if M > m.
Setting t = (n — 4))/2, z = pei(n+^)/2 g D in (5) and using univalency of Fr,0 for 0 < r < 1, we get
£
k=1
(ak zk + ei0 bk zk )rk-1
sin kt sin t
> A(r, a),
:i3)
where A(r, a) is defined as in (6). In order to find a lower bound for lPn,m,r,a(z)l, we need to find an upper bound for
|R
n, m, r, a
(z)|
£
k=n+1
akrk-1zk
sin kt sin t
+ £
k=m+1
(bk r z k)
sin kt sin t
From Theorem 2, it follows that
| Rn, m, r, a (z) | ^J] [kAk (a)rk-^ + J] k=n+1 k=m+1
= Rn(r, a) + Tm(r,a),
k(k — 1)
(k — 2a)
where Ak (a) is defined as in (9). From (13) and (14), we get
Ak(a)r
k-1
:i4)
n, m, r, a
(z)| ^ A(r, a) — Rn(r, a) — Tm(r, a) = ^(n, m, r, a).
The quantity Pn,m,r,a(z) = 0 holds for all z GD\{0}, whenever ip(n, m, r, a) defined by (12) is positive. This gives ^(n, m, r,a) > 0 for all r G (0, rn,m), where rn,m is the positive root of the equation ^>(n,m,r,a) = 0, which lies in the interval (0,1). This observation proves that Fr d is univalent in D for all 9 G [0, 2n]; this implies univalency of sn,m(f; 9) (see [10]) in the disk |z| < rn,m for all 9 G [0, 2n]. This completes the proof of the first part of the theorem.
From the above discussion, it is clear that rn,m ^ ri,i, where l = min{n,m} ^ 2. Next, we shall consider the special case m = n
and determine the lower bound for rn,n with a certain restriction on n and special choices of a.
Let us consider the case a = 1/2. In this case, the sufficient condition (12) for the univalence of sn,n(f; 0) reduces to
^(n, n, r, 1/2) = A(r, 1/2) - R„(r, 1/2) - Tra(r, 1/2) = = (1 -r2) (2'0g(1+r) - r)2-2 £ r- * -(15)
V r J fc=n+i (1 +r)2 (1 - r)
From the fact that lim ^(n,n,r, 1/2) > 0 for all r such that 0 < r < 1,
it is clear that the radius of univalence rn,n ^ 1 as n ^ ro. Setting r = 1 - x/n, where x = o(n), we see that
*(n,n,r, 1/2) =(1 -(lr)! > 0.
whenever 0 < t(x, n) < 1, where
t(x,n) = 2e-x f— - 1 x
Now, we shall prove that rn,n > 1 - Yn/n and we shall explicitly determine the value Yn for large values of n. In order to prove that, it is sufficient to prove that t(x, n) is a decreasing function in x, whenever
Yn ^ x ^ n, 0 < t(7n, n) < 1 and t(n, n) > 0.
Since ext(x,n) = O(n2), we may set Yn = 2 log n. It is easy to see that 1 - Yn/n > 0 and increases for all n ^ 3. For n ^ 3, we shall prove that
' n, n
> 1 - Yn/n.
Fix an integer n ^ 3 and consider an x G [Yn, n]; we prove that t(x, n) is a decreasing function with respect to x. From the definition of t(x,n), it is clear that it is a product of two positive decreasing functions. Hence, t(x, n) is a decreasing function with respect to x for every fixed integer n ^ 3. Further, it is easy to see that t(n, n) < 1 for all n ^ 3. In order to complete the proof, it is enough to show that t(Yn, n) < 1 for all
n ^ 3. Note that t(Yn,n) = ^--1 ) = 2 (--1 ) < 1 for
n2 log n log n n
all n ^ 3. The proof is complete. □
Remark 2. From Theorem B, we could see that whenever f G F(-1/2), sn,m(f; 0) is univalent in the disk |z| < rn,m, where
rn,m > rM = 1 - (4logl - 2log(logl))/l
for l = min{m, n} ^ 7.
Generally, it is possible to estimate lower bounds for rn,m for functions f G F(a) for a G [-1/2,1); however, if we consider the computation for finding the lower bound on rn,m in Theorem 3 and Theorem B, the difficulty in handling the general case a G [-1/2,1) becomes clear. Nevertheless, we can compare the variations on the radius of univalence of the partial sums, when we consider functions f G F(a) for various values of a G [-1/2,1). We demonstrate this in the following Corollary.
Corollary. Suppose that f G F(a) for some a G [-1/2,1). Then the value of n for which sn,n(f; 9) is univalent in the disk |z| < p for all values of 9 G [0, 2n] is given in Table 1:
|z| < p 1 a = — a 2 1 a = — a 4 a = 0 1 a = 4 1 a = 2 3 a = 4
1/4 n ^ 2 n ^ 2 n ^ 2 n ^ 2 n ^ 2 n ^ 2
1/2 n ^ 12 n ^ 10 n ^ 8 n ^ 6 n ^ 4 n ^ 3
3/4 n ^ 43 n ^ 36 n ^ 29 n ^ 22 n ^ 16 n ^ 9
9/10 n ^ 160 n ^ 135 n ^ 111 n ^ 86 n ^ 61 n ^ 35
Table 1: Values of n for which sn,n(f; 9) is univalent in the disk |z| < p
Proof. The lower bound on p, corresponding to the value n = 2, follows from the result [10, Proposition 2.1] and the result of Szego [23]. All the other lower bounds have been estimated with the help of the Mathematica software by estimating ip(n,n,r,a) = 0 in the equation (12), corresponding to the values of r = p, fixing a = -1/2, -1/4, 0,1/4,1/2, 3/4. □
Acknowledgment. The author thanks Prof. S. Ponnusamy for fruitful discussion and comments on this manuscript. The author also thanks the referees for their comments and suggestions, which helped the author to improve the results in this article.
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Received February 24, 2018.
In revised form, November 28, 2018.
Accepted November 28, 2018.
Published online December 13, 2018.
Indian Institute of Technology Ropar,
Nangal Road, Rupnagar, Punjab, 140001, India.
Email: [email protected]; [email protected]
