HJM, Vol. 47, No. 4, 2021

HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 47, No. 4, 2021

Editors:  D. Bao (San Francisco, SFSU), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), M. Dugas (Baylor), M. Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), S. Harvey (Rice), A. Haynes (Houston), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station),  M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and K. Kaiser (Houston)

 Houston Journal of Mathematics


Contents

S. Rahrovi, Department of Mathematics, University of Bonab, Bonab, Iran (sarahrovi@gmail.com), H. Piri, Department of Mathematics, University of Bonab, Bonab, Iran (h.piri@bonabu.ac.ir), and R. Kargar, Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland (rakarg@utu.fi, rkargar1983@gmail.com).
The behavior of starlike functions exterior of parabola, pp. 723–743.

ABSTRACT. In the present paper we study functions f which are analytic and normalized in the open unit disc such that satisfy the following subordination
( ′ ) 2 zf(z)− 1 ≺ -z −-tz (t ∈ [0,1]). f (z) (1 − z)2

Some properties of these functions including the radius of starlikeness and convexity, majoriziation problem, initial coefficient estimates, Fekete-Szegö problem and estimation of initial logarithmic coefficients and pre-Schwarzian norm are investigated.

 

Huanhuan Wei, School of Science, China University of Mining and Technology, Beijing 100083, P.R. China (1941522828@qq.com), Yuhua Li, School of Mathematics, Yunnan Normal University,Kunming 650500, P.R. China (liyuhua@ynnu.edu.cn), and Min Su, School of Mathematics, Yunnan Normal University, Kunming 650500, P.R. China (mathesumin@yahoo.com).
The relationship between two exponential sums sharing 0 IM, pp. 745–755.

ABSTRACT. The Shapiro’s conjecture states that if two non-constant exponential sums f(z) and g(z) have infinitely many zeros in common, then there exists a non-constant exponential sum h(z) with infinitely many zeros, such that h(z) is a common factor of f(z) and g(z) in the ring of exponential polynomials with constant coefficients. In this paper we prove that if two non-constant exponential sums f(z) and g(z) share 0 IM in ℂ, and f(z) has at least one zero, then the conclusion of the Shapiro’s conjecture is true.  

Ming Xu, School of Mathematical Sciences, Capital Normal University, Beijing 100048, P. R. China (mgmgmgxu@163.com).
Homogeneous Finsler sphere with constant flag curvature, pp. 757–771.

ABSTRACT. We prove that a homogeneous Finsler sphere with constant flag curvature K ≡ 1 and a prime closed geodesic of length 2π must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative method proving that a geodesic orbit Finsler sphere with K ≡ 1 must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with K ≡ 1. We prove that many geodesic properties for homogeneous Randers spheres with K ≡ 1 can be generalized to the non-Randers case.  

Chao Chen, School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, P.R. China (chenc225@163.com), Huibin Chen, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China (chenhuibin@njnu.edu.cn), and Zhiqi Chen, School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, P.R. China (chenzhiqi@nankai.edu.cn).
New invariant Einstein and Einstein-Randers metrics on certain homogeneous spaces arising from flag manifolds, pp. 773–794.

ABSTRACT. In this paper, we construct new invariant Einstein and Einstein-Randers metrics on certain homogeneous spaces G∕H of exceptional type, which are arising from flag manifolds G∕K(H ⊂ K) with two isotropy summands. By viewing these homogeneous spaces as total spaces over corresponding flag manifolds, we consider G-invariant metrics on G∕H determined by Ad(K)-invariant inner products on m′ = To(G∕H). By solving the homogeneous Einstein equations, we obtain many invariant Einstein metrics on G∕H. Furthermore, based on the results in Riemannian case, we construct new examples of Einstein-Randers metrics on G∕H.  

Xiaohuan Mo, Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China (moxh@math.pku.edu.cn), and Hongzhen Zhang, Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China (zhz_93@163.com).
Finsler (or spray) manifolds with S-sprays, pp. 795–809.

ABSTRACT. The S-spray plays an important role in discussing Finsler/spray measure spaces. In this paper, we study the geometry of S-sprays on a manifold. We obtain a lot of examples of S-sprays. We find necessary and sufficient conditions for the spray induced by a Randers/(α,β)-metric to be an S-spray. Finally, we give a global rigidity result with respect to complete S-spray generalized a theorem previously only known in the case of projectively Ricci flat sprays.  

Saminathan Ponnusamy, Department of Mathematics, Indian Institute of Technology Mad-ras, Chennai-600 036, India (samy@iitm.ac.in), Ramakrishnan Vijayakumar, Department of Mathematics, Indian Institute of Technology Mad-ras, Chennai-600 036, India (mathesvijay8@gmail.com), and Karl-Joachim Wirths, Institut für Analysis und Algebra, TU Braunschweig, 38106 Braunschweig, Germany (kjwirths@tu-bs.de).
Modifications of Bohr’s inequality in various settings, pp. 811–835.

ABSTRACT. The concept of Bohr radius for the class of bounded analytic functions was introduced by Harald Bohr in 1914. His initial result received greater interest and was sharpened-refined-generalized by several mathematicians in various settings–which is now called Bohr phenomenon. Various generalization of Bohr’s classical theorem is now an active area of research and has been a source of investigation in numerous other function spaces and including holomorphic functions of several complex variables. Recently, a new generalization of Bohr’s ideas was introduced and investigated by Kayumov et al. In this note, we investigate and refine generalized Bohr’s inequality for the class of quasi-subordinations.  

Ermin Wang, School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China (wem0913@sina.com), and Zhenghua Xu, School of Mathematics, Hefei University of Technology, Hefei, Anhui 230601, China (zhxu@hfut.edu.cn).
Hankel operators between classical Fock spaces, pp. 837–852.

ABSTRACT. In this paper, for all possible 1 ≤ p,q < ∞, we characterize these complex-valued symbols f for which the induced Hankel operators Hf are bounded (or compact) from Fαp to Lαq. In addition, we discuss the Schatten class membership of Hf on the Hilbert space Fα2.  

Zhi-Bo Huang, School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China (huangzhibo@scnu.edu.cn), Ilpo Laine, Department of Physics and Mathematics, University of Eastern Finland, P.O.Box 111, FI-80101 Joensuu, Finland (ilpo.laine@uef.fi), and Min-Wei Luo, School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China (2019021673@m.scnu.edu.cn).
Growth of solutions to higher order differential equations with Mittag-Leffler coefficients, pp. 853–865.

ABSTRACT. The classical problem of finding conditions on the entire coefficients Aj (j = 0,1,⋅⋅⋅,k − 1) ensuring that all nontrivial solutions to higher order differential equations f(k) + Ak−1f(k−1) + ⋅⋅⋅ + A1(z)f′ + A0(z)f = 0 are of infinite lower order is being discussed in this paper. In particular, we assume that the coefficients (or most of them) are Mittag-Leffler functions.  

Yanni Chen, School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, China (yanni.chen@snnu.edu.cn), Junsheng Fang, School of Mathematics, Hebei Normal University, Shijiazhuang, China (Junshengfang@gmail.com), and Don Hadwin, Mathematics Department, University of New Hampshire, Durham, NH (don@unh.edu).
Vector-valued Lebesgue and Hardy spaces for symmetric norms over compact groups, pp. 867–910.

ABSTRACT. In this paper, we study gauge norms on probability spaces and their associated Lebesgue spaces, both in the scalar and vector-valued cases. We consider norms that are symmetric with respect to groups of measure-preserving transformations, and we show that such a group is ergodic if and only if every symmetric gauge norm dominates ∥∥1. When the probability space is a compact group G with Haar measure μ, we study convolution of Banach algebra-valued functions in Lebesgue spaces. When G is Abelian and its dual group is linearly ordered, we study the associated Hardy spaces. When G = T, we characterize the closed densely defined operators on Hα(T) affiliated with H∞(T).  

Gianluigi Manzo, Dipartimento di Matematica e Applicazioni ”R. Caccioppoli”, Università degli studi di Napoli ”Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy (gianluigi.manzo@unina.it).
Some characterizations of a family of spaces defined by means of oscillations, pp. 911–934.

ABSTRACT. We study the family of spaces BMO(s) ⊂ BMO(1) = BMO, which can be defined by uniform EXPs conditions on oscillations. These spaces are dual of H(s)1, which can be defined by atomic decomposition and are themselves duals of a ”little-o” space V MO(s) ⊂ BMO(s). We show results on duality, distance formulas and M-ideality for these spaces using properties of o-O type structures. We also show a Sobolev-type embeddings involving Lorentz spaces Ln,q.  

Süleyman Önal, Middle East Technical University, Department of Mathematics, 06531 Ankara, Turkey (osul@metu.edu.tr), and Çetin Vural, Gazi Üniversitesi, Fen Fakültesi, Matematik Bölümü, 06500 Teknikokullar, Ankara, Turkey (cvural@gazi.edu.tr).
Completeness properties in topological spaces having a pair-base, pp. 935–948.

ABSTRACT. We introduce the completeness of a pair-base and study the topological spaces having such a base. We investigate, among the others, Čech-complete spaces and subcompact spaces have a complete pair-base, and we prove that if a topological space X has a complete pair-base then X is domain representable. We establish that a paracompact p-space X must be Čech-complete if it has a countable complete pair-base. We also show that the property of having a complete pair-base is preserving under retractions.  

Javier Casas-de la Rosa, Department of Mathematics and Statistics, York University, 4700 Keele St. Toronto, ON M3J 1P3 Canada (olimpico.25@hotmail.com), Iván Martínez-Ruiz, Facultad de Ciencias Físico Matemáticas, Benémerita Universidad Autónoma de Puebla, Ave. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue. C.P. 72570, México (imartinez@fcfm.buap.mx), and Alejandro Ramírez-Páramo Facultad de Ciencias de la Electrónica, Benémerita Universidad Autónoma de Puebla, Ave. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue. C.P. 72570, México (alejandro.ramirez@correo.buap.mx).
Star versions of the Menger property on hyperspaces, pp. 949–960.

ABSTRACT. We employ the notion of πF(Δ)–network to define the combinatorial principles FELLM(ΠF(Δ),ΠF(Δ)) and FELLM∗(ΠF(Δ)),ΠF(Δ)), which will be applied to characterize the spaces X whose hyperspace, endowed with the Fell topology, satisfies the SSM condition and the SM condition. We use the selection principle SSΔ∗(O,O) to characterize the SSM property for the spaces K(X), F(X) and [X]1, endowed with the lower Vietoris topology. Finally, we use the notion of Δ-moving-off family, which generalizes the one of moving-off family, and we use it to characterize the Menger property for certain subspaces of CL(X).  

Juan Luis García Guirao Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena, C/ Paseo Alfonso XIII, 30203-Cartagena, Región de Murcia, Spain (juan.garcia@upct.es), Jaume Llibre Departament de Matemàtiques. Universitat Autònoma de Barcelona, Bellaterra, 08193-Barcelona, Catalonia, Spain (jllibre@mat.uab.cat) Wei Gao School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China (gaowei@ynnu.edu.cn).
C1 self–maps on some compact manifolds with infinitely many hyperbolic periodic orbits, pp. 961–974.

ABSTRACT. The aim of the present work is to provide sufficient conditions for having infinitely many periodic points for C1 self–maps having all their periodic orbits hyperbolic and defined on a compact manifold without boundary. The tool used for proving our results is the Lefschetz fixed point theory.