Editors: D. Bao (San Francisco, SFSU), S. Berhanu (Temple), D. Blecher
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M. Gehrke (CNRS),
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M. Ru (Houston), S. W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and A. Haynes (Houston)
Houston Journal of Mathematics
Contents
Geng-chen Wang, School of Mathematics, Nanjing University, Nanjing 210093, China
(gengchenwang@126.com), and Liang-wen Liao, School of Mathematics, Nanjing
University, Nanjing 210093, China (maliao@nju.edu.cn).
On the entire solutions of a certain type of nonlinear differential equations and extended
Duffing equation, pp. 1–14.
ABSTRACT. In this paper, by using Nevanlinna theory, we obtain the explicit forms of entire solutions of the following differential equation
where n ≥ 3, p(z) is a polynomial, Q(z,f) is an algebraic differential polynomial in f of degree m ≤ n− 2 with polynomials as its coefficients, and a(z) is a non-constant entire function. Moreover, we study the extended Duffing differential equation by using our result and obtain its entire solutions.
Abhijit Banerjee, Department of Mathematics, University of Kalyani, West Bengal
741235, India. (abanerjee_kal@yahoo.co.in), Sujoy Majumder, Department
of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India.
(sm05math@gmail.com, sjm@raiganjuniversity.ac.in), and Nabadwip Sarkar, Amity
School of Applied Sciences, Amity University Mumbai, Panvel, Navi Mumbai,
Maharashtra-410206, India. (naba.iitbmath@gmail.com, nsarkar@mum.amity.edu).
Uniqueness of first derivatives and differences in meromorphic functions and the
characterization of entire function periodicity, pp. 15–38.
ABSTRACT. The objective of the paper is twofold. The first objective is to study the
uniqueness problem of meromorphic function f(z) when f(1)(z) shares two distinct finite
values a1, a2 and ∞ CM with Δcf(z). In this context, we provide a result that
resolves the open problem posed by Qi et al. [Comput. Methods Funct. Theory, 18
(2018), 567-582] for the case when hyper order of the function is less than ∞.
The second objective is to establish sufficient conditions for the periodicity
of transcendental entire functions. In this direction, we obtain a result that
affirms the question raised by Wei et al. [Anal. Math., 47 (2021), 695-708.]
Molla Basir Ahamed, Department of Mathematics, Jadavpur University,
Kolkata-700032, West Bengal, India (mbahamed.math@jadavpuruniversity.in), Partha
Pratim Roy, Department of Mathematics, Jadavpur University, Kolkata-700032,
West Bengal, India (pproy.math.rs@jadavpuruniversity.in), Rajesh Hossain,
Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India (rajesh1998hossain@gmail.com), and Sabir Ahammed, Department of
Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India
(sabirahammed517@gmail.com).
The Bohr inequality and the Bohr-Rogosinski inequality for stable harmonic mappings,
pp. 39–62.
ABSTRACT. Bohr’s original theorem and its subsequent generalizations remain
active fields of study for different classes of functions. Since not every class of
functions has Bohr phenomenon, it is of interest to know which class of functions
has Bohr phenomenon. Let ℋ0 denote the set of all sense-preserving harmonic
mappings f = h + g in the unit disk 𝔻, normalized with h(0) = g(0) = g′(0) = 0
and h′(0) = 1. In this paper we study Bohr inequality and Bohr-Rogosinski
inequality in view of distance formulation for certain sub-classes of ℋ0, known
as stable harmonic mappings. Also we established several sharp inequalities.
Toshiki Ambo, Department of Mathematics, Saitama University, Saitama, Japan
(t.ambo.993@ms.saitama-u.ac.jp), and Yuki Takahashi, Department of Mathematics,
Saitama University, Saitama, Japan (ytakahas@rimath.saitama-u.ac.jp).
On sums of f-transformed semibounded Cantor sets, pp. 63–75.
ABSTRACT. For monotonic C1 functions f : (0,∞) → ℝ with limx→+0f(x) = −∞, we
obtain estimates in terms of thickness to guarantee that the set f(K1 ∖{0}) + f(K2 ∖{0}) is
a half-line for all Cantor sets K1,K2 ⊂ ℝ with minK1 = minK2 = 0.
Ryo Toyota, Department of Mathematics, Texas A&M University, TX, USA
(ryo-toyota@tamu.edu), and Zhiyuan Yang, Department of Mathematics, Texas A&M
University, TX, USA (zhiyuanyang@tamu.edu).
An operator-valued Haagerup inequality for hyperbolic groups, pp. 77–86.
ABSTRACT. We study an operator-valued generalization of the Haagerup inequality for
hyperbolic groups. In 1978, U. Haagerup showed that if f ∈ ℂ[𝔽r] is supported on the
k-sphere Sk = {x ∈ 𝔽r : ℓ(x) = k}, then we have 
B(ℓ2(𝔽r)) ≤ (k + 1)∥f∥2.
An operator-valued generalization of it was initiated by U. Haagerup and G. Pisier. One
of the most complete form was given by A. Buchholz, where the ℓ2-norm in the
original inequality was replaced by k + 1 different matrix norms associated to
word decompositions (this type of inequality is also called Khintchine-type
inequality). We provide a generalization of Buchholz’s result for hyperbolic groups.
Akitake Han, (han.akitake@lemon.plala.or.jp), and Kazuhiro Kawamura,
Department of Mathematics, University of Tsukuba, Tsukuba Ibaraki 305-8571, Japan
(kawamura@math.tsukuba.ac.jp).
Hypercyclicity of weighted backward shifts and branching growth rates of directed trees,
pp. 87–117.
ABSTRACT. We study hypercyclic/mixing weighted backward shift operators on
ℓp- and c0-sequence spaces over locally finite directed trees in connection with
combinatorics of the trees. We introduce a couple of invariants of a tree that reflect
branching growth and apply them to study the operator on the basis of the
characterization theorem due to Grosse-Erdmann and Papathanasiou (Dynamics of
weighted shifts on directed trees, Indiana Univ. Math. J. 72 (2023), 263-299).
Guodong Hua, School of Mathematics and Statistics, Weinan Normal University,
Weinan, Shaaxi Province, China 714099; Research Institute of Qindong
Mathematics, Weinan Normal University, Weinan, Shaaxi Province, China 714099
(gdhuasdu@163.com).
On higher power moments of general divisor problem of Hecke eigenvalues over certain
sparse sequences of positive integers, pp. 119–146.
ABSTRACT. Let f and g be two distinct primitive normalized holomorphic cuspidal
Hecke eigenforms of even integral weights κ1 and κ2 for the full modular group
Γ = SL(2, ℤ), and denote its n-th Fourier coefficients by λf(n) and λg(n), respectively.
In this paper, we consider the general divisor problem for λf2ℓ1(n)λg2ℓ2(n) over certain
multivariate polynomials of four variables, where ℓ1,ℓ2 ≥ 1 are any given positive
integers. Precisely, we establish the asymptotic formulae for the general divisor sums
involving λf2ℓ1(n)λg2ℓ2(n), which are supported at the a sum of triangular numbers with
certain positive coefficients. The results generalizes the previous work in this direction.
Ting Wang, Department of Mathematics, Shanghai University, Shanghai 200444, China
(wangting0729@shu.edu.cn), Qiong Wu, Department of Mathematics, Shanghai
University, Shanghai 200444, China (15722996870@shu.edu.cn), and Weili Yao,
Department of Mathematics, Shanghai University, Shanghai 200444, China;
Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China
(yaoweili@shu.edu.cn).
Mixed average behavior of coefficients of the Dedekind zeta-function, pp. 147–161.
ABSTRACT. Let 𝕂 be a quadratic field with discriminant D, and let ζ𝕂(s) be its
Dedekind zeta-function with coefficients a𝕂(n). This paper investigates the mixed
average behavior of a𝕂(n) over sparse sets in short intervals, and obtains an interesting
asymptotic formula for it.
Li-Hong Xie, School of Mathematics and Computational Science, Wuyi University,
Jiangmen, Guangdong, 529000, P.R. China (yunli198282@126.com), Hanyi Ni, School
of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong,
529000, P.R. China (931479204@qq.com), Yi-Ting Wang, School of Mathematics
and Computational Science, Wuyi University, Jiangmen, Guangdong, 529000,
P.R. China (2205616228@qq.com), and Ying-Ying Jin, General Education
Department, Guangzhou Polytechnic University, Guangzhou 511483, P.R. China
(yingyjin@163.com).
Some characterizations of paratopological groups with a q-point, pp. 163–181.
ABSTRACT. Paratopological groups with a q-point are investigated. We mainly show
that: (1) A regular paratopological group G is of pointwise countable type iff for each
open neighborhood U of the neutral element e there is a closed compact neutral
subgroup H ⊆ U of G such that the quotient space G∕H is regular, quasi-metrizable and
the canonical quotient mapping p : G → G∕H is perfect; (2) a regular paratopological
group G with Sm(G) ≤ ω is a q-space iff for each open neighborhood U of the neutral
element e there is a closed countably compact neutral subgroup H ⊆ U of G such that
the quotient space G∕H is regular, quasi-metrizable and the canonical quotient mapping
p : G → G∕H is quasi-perfect; (3) a regular paratopological group G is a strong
q-space iff each open neighborhood U of e contains a closed sequentially compact
neutral subgroup H of countable character, which is a strong q-sequence at each
x ∈ H in G, such that the quotient space G∕H is regular, quasi-metrizable and
the canonical quotient mapping p : G → G∕H is strongly sequential-perfect.
Jiajia Yang, School of mathematics and statistics, Minnan Normal University,
Zhangzhou 363000, P. R. China (yjj8030@163.com), Jiamin He, School of
mathematics and statistics, Minnan Normal University, Zhang-zhou 363000, P. R.
China (hjm1492539828@163.com), and Fucai Lin, 1. School of mathematics
and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China;
2. Fujian Key Laboratory of Granular Computing and Application, Minnan
Normal University, Zhangzhou 363000, P. R. China (linfucai2008@aliyun.com;
linfucai@mnnu.edu.cn).
The existence of suitable sets in locally compact strongly topological gyrogroups, pp.
183–194.
ABSTRACT. A subset S of a topological gyrogroup G is said to be a suitable set for G
if S is discrete, the gyrogroup generated by S is dense in G, and S ∪{0} is closed in G,
where 0 is the identity element of G. In this paper, it is proved that every locally
compact strongly topological gyrogroup has a suitable set, which gives an affirmative
answer to a question posed by F. Lin et al.
Zhengmao He, School of Sciences, Southwest Petroleum University, Chengdu 610500,
Sichuan, China (hezhengmaomath@163.com).
The sobriety of Fréchet-Urysohn spaces and Scott topology of the lattice of open sets,
pp. 195–212.
ABSTRACT. In this paper, we investigate the sobriety of weakly first-count-able spaces and give some sufficient conditions such that the Scott topologies of the open set lattices are sober. The main results are:
(1) Let P and Q be two posets. If ΣP × ΣQ is a Fréchet-Urysohn space, then Σ(P × Q) = ΣP × ΣQ.
(2) For every ω-well-filtered coherent d-space X, if X ×X is a Fréchet-Urysohn space, then X is sober;
(3) Let X be an ω-well-filtered space (ω-d-space). If Xω (Xd) is a Fréchet-Urysohn space, then X is a well-filtered space (d-space).
(4) For every ω-type P-space or consonant Wilker space X, Σ𝒪(X) is sober.
Chetana Visave, Department of Mathematics, University of Mumbai
(visavechetana94@gmail.com), and Rajendra Deore, Department of Mathematics,
University of Mumbai (rpdeore@gmail.com).
Corrigendum to “Power Graph of Some Finite and Infinite Groups”, pp. 213–223.
ABSTRACT. This is a corrigendum to our research paper titled “Power Graph of Some Finite and Infinite Groups,” HJM 51-3 (2025), 425–451. In literature the power graph of a finite group is defined in two different ways. We observed that both the definitions agree for the power graphs of finite groups. But in case of power graphs of infinite groups, the two definitions led to two different structures of power graphs. In this corrigendum, we have given the construction of 𝒫(ℤ) with respect to both the definitions and investigate the properties of 𝒫(ℤ) with respect to both the constructions. We deeply regret a minor error in the construction of a power graph 𝒫(ℤ) of an infinite cyclic group ℤ. We also obtain a few new results in this context.