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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 35, Num. 1 (2020), 35 – 42

doi:10.17398/2605-5686.35.1.35

Available online January 7, 2020

On H3(1) Hankel determinant for certain
subclass of analytic functions

D. Vamshee Krishna 1,@, D. Shalini 2

1 Department of Mathematics, GIS, GITAM University
Visakhapatnam- 530 045, A.P., India

2 Department of Mathematics, Dr. B. R. Ambedkar University
Srikakulam- 532 410, A.P., India

vamsheekrishna1972@gmail.com , shaliniraj1005@gmail.com

Received February 21, 2019 Presented by Manuel Maestre
Accepted September 3, 2019

Abstract: The objective of this paper is to obtain an upper bound to Hankel determinant of third

order for any function f, when it belongs to certain subclass of analytic functions, defined on the
open unit disc in the complex plane.

Key words: Analytic function, upper bound, third Hankel determinant, positive real function.

AMS Subject Class. (2010): 30C45, 30C50.

1. Introduction

Let A denotes the class of analytic functions f of the form

f(z) = z +
∞∑
n=2

anz
n (1.1)

in the open unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting
of univalent functions. In 1985, Louis de Branges de Bourcia proved the
Bieberbach conjecture also called as Coefficient conjecture, which states that
for a univalent function its nth- Taylor’s coefficient is bounded by n (see [4]).
The bounds for the coefficients of these functions give information about their
geometric properties. For example, the nth-coefficient gives information about
the area where as the second coefficient of functions in the family S yields
the growth and distortion properties of the function. A typical problem in
geometric function theory is to study a functional made up of combinations
of the coefficients of the original function. The Hankel determinant of f for

@ Corresponding author

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.35.1.35
mailto:vamsheekrishna1972@gmail.com
mailto:shaliniraj1005@gmail.com
mailto:vamsheekrishna1972@gmail.com
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


36 d. vamshee krishna, d. shalini

q ≥ 1 and n ≥ 1 was defined by Pommerenke [20], which has been investigated
by many authors, as follows.

Hq(n) =

an an+1 · · · an+q−1
an+1 an+2 · · · an+q

...
...

...
...

an+q−1 an+q · · · an+2q−2

. (1.2)

It is worth of citing some of them. Ehrenborg [7] studied the Hankel deter-
minant of exponential polynomials. Noor [18] determined the rate of growth
of Hq(n) as n → ∞ for the functions in S with bounded boundary rota-
tion. The Hankel transform of an integer sequence and some of its properties
were discussed by Layman (see [13]). It is observed that H2(1), the Fekete-
Szegö functional is the classical problem settled by Fekete-Szegö [8] is to find
for each λ ∈ [0, 1], the maximum value of the coefficient functional, defined
by φλ(f) := |a3 − λa22| over the class S and was proved by using Loewner
method. Ali [1] found sharp bounds on the first four coefficients and sharp
estimate for the Fekete-Szegö functional |γ3 − tγ22|, where t is real, for the in-
verse function of f defined as f−1(w) = w +

∑∞
n=2 γnw

n when f−1 ∈ S̃T(α),
the class of strongly starlike functions of order α (0 < α ≤ 1). In recent years,
the research on Hankel determinants has focused on the estimation of |H2(2)|,
where

H2(2) =
a2 a3
a3 a4

= a2a4 −a23,

known as the second Hankel determinant obtained for q = 2 and n = 2 in
(1.2). Many authors obtained an upper bound to the functional |a2a4−a23| for
various subclasses of univalent and multivalent analytic functions. It is worth
citing a few of them. The exact (sharp) estimates of |H2(2)| for the subclasses
of S namely, bounded turning, starlike and convex functions denoted by R,
S∗ and K respectively in the open unit disc E, that is, functions satisfying
the conditions Ref ′(z) > 0, Re

{
zf′(z)
f(z)

}
> 0 and Re

{
1 +

zf′′(z)
f′(z)

}
> 0 were

proved by Janteng et al. [11, 10] and determined the bounds as 4/9, 1 and
1/8 respectively. For the class S∗(ψ) of Ma-Minda starlike functions, the
exact bound of the second Hankel determinant was obtained by Lee et al.
[15]. Choosing q = 2 and n = p + 1 in (1.2), we obtain the second Hankel
determinant for the p-valent function (see [24]), as follows.

H2(p + 1) =
ap+1 ap+2
ap+2 ap+3

= ap+1ap+3 −a2p+2,



on H3(1) hankel determinant 37

The case q = 3 appears to be much more difficult than the case q = 2.
Very few papers have been devoted to the third order Hankel determinant
denoted by H3(1), obtained for q = 3 and n = 1 in (1.2), also called as Hankel
determinant of third kind, namely

H3(1) =
a1 a2 a3
a2 a3 a4
a3 a4 a5

(a1 = 1).

Expanding the determinant, we have

H3(1) = a1(a3a5 −a24) + a2(a3a4 −a2a5) + a3(a2a4 −a
2
3), (1.3)

equivalently
H3(1) = H2(3) + a2J2 + a3H2(2),

where J2 = (a3a4 −a2a5) and H2(3) = (a3a5 −a24).

Babalola [2] is the first one, who tried to estimate an upper bound for
|H3(1)| for the classes R, S∗ and K. As a result of this paper, Raza and
Malik [22] obtained an upper bound to the third Hankel determinant for a
class of analytic functions related with lemniscate of Bernoulli. Sudharsan
et al. [23] derived an upper bound to the third kind Hankel determinant
for a subclass of analytic functions. Bansal et al. [3] improved the upper
bound for |H3(1)| for some of the classes estimated by Babalola [2] to some
extent. Recently, Zaprawa [25] improved all the results obtained by Babalola
[2]. Further, Orhan and Zaprawa [19] obtained an upper bound to the third
kind Hankel determinant for the classes S∗ and K functions of order alpha.
Very recently, Kowalczyk et al. [12] calculated sharp upper bound to |H3(1)|
for the class of convex functions K and showed as |H3(1)| ≤ 4135, which is far
better than the bound obtained by Zaprawa [25]. Lecko et al. [14] determined
sharp bound to the third order Hankel determinant for starlike functions of
order 1/2. Motivated by the results obtained by different authors mentioned
above and who are working in this direction (see [5]), in this paper, we are
making an attempt to obtain an upper bound to the functional |H3(1)| for
the function f belonging to the class, defined as follows.

Definition 1.1. A function f(z) ∈ A is said to be in the class Q(α,β,γ)
with α, β > 0 and 0 ≤ γ < α + β ≤ 1, if it satisfies the condition that

Re

{
α
f(z)

z
+ βf ′(z)

}
≥ γ, z ∈ E. (1.4)



38 d. vamshee krishna, d. shalini

This class was considered and studied by Zhi- Gang Wang et al. [26].

In obtaining our results, we require a few sharp estimates in the form of
lemmas valid for functions with positive real part.

Let P denotes the class of functions consisting of g, such that

g(z) = 1 + c1z + c2z
2 + c3z

3 + · · · = 1 +
∞∑
n=1

cnz
n, (1.5)

which are analytic in E and Reg(z) > 0 for z ∈ E. Here g is called the
Caratheodòry function [6].

Lemma 1.2. ([9]) If g ∈ P, then the sharp estimate |ck −µckcn−k| ≤ 2,
holds for n,k ∈ N = {1, 2, 3, . . .}, with n > k and µ ∈ [0, 1].

Lemma 1.3. ([17]) If g ∈ P, then the sharp estimate |ck − ckcn−k| ≤ 2,
holds for n,k ∈ N, with n > k.

Lemma 1.4. ([21]) If g ∈ P then |ck| ≤ 2, for each k ≥ 1 and the inequal-
ity is sharp for the function g(z) = 1+z

1−z , z ∈ E.

In order to obtain our result, we refer to the classical method devised by
Libera and Zlotkiewicz [16], used by several authors.

2. Main result

Theorem 2.1. If f(z) = z +
∑∞

n=2 anz
n ∈ Q(α,β,γ), (α, β > 0 and

0 ≤ γ < α + β ≤ 1) then

|H3(1)| ≤ 4t21
[
k1α

6 + k2α
5 + k3α

4β + k4α
3β2 + k5α

2β3 + k6αβ
4 + k7β

5

(α + 2β)2(α + 3β)3(α + 4β)2(α + 5β)

]
,

where k1 = 2, k2 = 2(18β + 1), k3 = 2(132β + 15), k4 = 2(511β + 87),
k5 = (2179β+490), k6 = 12(203β+56), k7 = 12(93β+30) and t1 = (α+β−γ).

Proof. Let f(z) = z+
∑∞

n=2 anz
n ∈ Q(α,β,γ). By virtue of Definition 1.1,

there exists an analytic function g ∈ P in the open unit disc E with g(0) = 1
and Re{g(z)} > 0 such that

1

α + β −γ

{
α
f(z)

z
+ βf ′(z) −γ

}
= g(z) (2.1)



on H3(1) hankel determinant 39

Using the series representation for f and g in (2.1), upon simplification, we
obtain

∞∑
n=2

(α + nβ)anz
n−2 = (α + β −γ)

∞∑
n=1

cnz
n−1. (2.2)

The coefficient of zt−2, where t is an integer with t ≥ 2 in (2.2) is given by

at =
(α + β −γ)ct−1

(α + tβ)
, with t ≥ 2. (2.3)

Substituting the values of a2, a3, a4 and a5 from (2.3) in the functional given
in (1.3), it simplifies to

H3(1) = (α + β −γ)2
[

c2c4
(α + 3β)(α + 5β)

−
(α + β −γ)c32

(α + 3β)3
−

c23
(α + 4β)2

−
(α + β −γ)c21c4

(α + 2β)2(α + 5β)
+

2(α + β −γ)c1c2c3
(α + 2β)(α + 3β)(α + 4β)

]
.

(2.4)

On grouping the terms in the expression (2.4), in order to apply the lemmas,
we have

H3(1) = t
2
1

[
c4(c2 − t1c21)

(α + 2β)2(α + 5β)
−

c3
(α + 4β)2

{
c3 −

t1(α + 4β)c1c2
(α + 2β)(α + 3β)

}
(2.5)

+
c2(c4 − t1c22)

(α + 3β)3
−

c2
(α + 3β)(α + 4β)2

{
c4 −

t1(α + 4β)c1c3
(α + 2β)(α + 4β)

}

+
(d1α

6 + d2α
5 + d3α

4β + d4α
3β2 + d5α

2β3 + d6αβ
4 + d7β

5)c2c4
(α + 2β)2(α + 3β)3(α + 4β)2(α + 5β)

]
,

with d1 = 1, d2 = (18β − 1), d3 = (133β − 19), d4 = 4(129β − 35), d5 =
2(554β − 249), d6 = 8(156β − 107), d7 = 4(144β − 143) and t1 = (α + β −γ).
On applying the triangle inequality in (2.5), we have∣∣∣H3(1)∣∣∣ ≤t21[ |c4||(c2 − t1c21)|(α + 2β)2(α + 5β) + |c3|(α + 4β)2

∣∣∣∣c3 − t1(α + 4β)c1c2(α + 2β)(α + 3β)
∣∣∣∣

+
|c2||(c4 − t1c22)|

(α + 3β)3
+

|c2|
(α + 3β)(α + 4β)2

∣∣∣∣c4 − t1(α + 4β)c1c3(α + 2β)(α + 4β)
∣∣∣∣ (2.6)

+
|d1α6 + d2α5 + d3α4β + d4α3β2 + d5α2β3 + d6αβ4 + d7β5||c2||c4|

(α + 2β)2(α + 3β)3(α + 4β)2(α + 5β)

]
.



40 d. vamshee krishna, d. shalini

Upon using the lemmas given in (1.2), (1.3) and (1.4) in the inequality (2.6),
it simplifies to

|H3(1)|

≤ 4t21
[
k1α

6 + k2α
5 + k3α

4β + k4α
3β2 + k5α

2β3 + k6αβ
4 + k7β

5

(α + 2β)2(α + 3β)3(α + 4β)2(α + 5β)

]
,

(2.7)

with k1 = 2, k2 = 2(18β + 1), k3 = 2(132β + 15), k4 = 2(511β + 87), k5 =
(2179β + 490), k6 = 12(203β + 56), k7 = 12(93β + 30) and t1 = (α + β −γ).
This completes the proof of the theorem.

Remark 2.2. For the values α = 1−σ, β = σ, γ = 0, so that (α+β−γ) = 1
in (2.7), we obtain

|H3(1)| ≤ 4
[

63σ6 + 312σ5 + 411σ4 + 414σ3 + 188σ2 + 44σ + 4

(1 + σ)2(1 + 2σ)3(1 + 3σ)2(1 + 4σ)

]
. (2.8)

Remark 2.3. Choosing σ = 1 in the expression (2.8), it coincides with the
result obtained by Zaprawa [25].

Acknowledgements

The authors are extremely grateful to the esteemed reviewers for
a careful reading of the manuscript and making valuable suggestions
leading to a better presentation of the paper.

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	Introduction
	Main result