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BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS Boyong Chen, Liangying Jiang Acta mathematica scientia,Series B    2024, 44 (3): 789-809.   DOI: 10.1007/s10473-024-0301-1 Abstract （208）            Save In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains. Reference | Related Articles | Metrics

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GENERALIZED COUNTING FUNCTIONS AND COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES OF DIRICHLET SERIES Min He, Maofa Wang, Jiale Chen Acta mathematica scientia,Series B    2025, 45 (2): 291-309.   DOI: 10.1007/s10473-025-0201-z Abstract （198）            Save In this paper, we study composition operators on weighted Bergman spaces of Dirichlet series. We first establish some Littlewood-type inequalities for generalized mean counting functions. Then we give sufficient conditions for a composition operator with zero characteristic to be bounded or compact on weighted Bergman spaces of Dirichlet series. The corresponding sufficient condition for compactness in the case of positive characteristics is also obtained. Reference | Related Articles | Metrics

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THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS∗ Hakho Hong Acta mathematica scientia,Series B    2023, 43 (2): 469-491.   DOI: 10.1007/s10473-023-0201-9 Abstract （196）      PDF       Save The aim of this work is to prove the existence for the global solution of a non-isothermal or non-isentropic model of capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), in the case of van der Waals gas. Under the small initial perturbation, the proof of the global existence is based on an elementary energy method using the continuation argument of local solution. Moreover, the uniqueness of global solutions and large time behavior of the density are given. It is one of the main difficulties that the pressure $p$ is not the increasing function of the density $\rho$. Reference | Related Articles | Metrics

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LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY* Zhongmin Qian, Xingcheng Xu Acta mathematica scientia,Series B    2024, 44 (5): 1609-1638.   DOI: 10.1007/s10473-024-0501-8 Abstract （158）            Save This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering. Reference | Related Articles | Metrics

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CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE* Xin Wei, Zhi-Ying Wen Acta mathematica scientia,Series B    2023, 43 (3): 981-993.   DOI: 10.1007/s10473-023-0301-6 Abstract （157）      PDF       Save Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves. Reference | Related Articles | Metrics

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THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL Han Luo Acta mathematica scientia,Series B    2024, 44 (4): 1189-1199.   DOI: 10.1007/s10473-024-0401-y Abstract （152）            Save In this paper, we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations. Under some suitable conditions, we give the gradient estimates of these maps and establish a Liouville type result. Reference | Related Articles | Metrics

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THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE* Huafei DI, Weijie Rong Acta mathematica scientia,Series B    2023, 43 (1): 324-348.   DOI: 10.1007/s10473-023-0118-3 Abstract （146）      PDF       Save This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation $u_{t}+(-\Delta)^{s_{1}} u_{t}+\beta(-\Delta)^{s_{2}}u=F(u,x,t)$ subject to random Gaussian white noise for initial and final data. Under the suitable assumptions $s_{1}$, $s_{2}$ and $\beta$, we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard, which are mainly driven by random noise. Moreover, we propose the Fourier truncation method for stabilizing the above ill-posed problems. We derive an error estimate between the exact solution and its regularized solution in an $\mathbb{E}\parallel\cdot\parallel^{2}_{H^{s_{2}}}$ norm, and give some numerical examples illustrating the effect of above method. Reference | Related Articles | Metrics

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THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHRÖDINGER EQUATION WITH MIXED DISPERSION* Tingjian Luo, Shijun Zheng, Shihui Zhu Acta mathematica scientia,Series B    2023, 43 (2): 539-563.   DOI: 10.1007/s10473-023-0205-5 Abstract （139）      PDF       Save In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by $Q_p$ the ground state for the BNLS with $\mu=0$, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu\in ( -\lambda_0, \infty)$ for some $\lambda_0=\lambda_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we prove the orbital stability on a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$. Reference | Related Articles | Metrics

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SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES* Manli Liu, Lingyun Gao, Shaomei Fang Acta mathematica scientia,Series B    2023, 43 (2): 821-838.   DOI: 10.1007/s10473-023-0218-0 Abstract （136）      PDF       Save In this paper, we mainly investigate the value distribution of meromorphic functions in $\mathbb{C}^m$ with its partial differential and uniqueness problem on meromorphic functions in $\mathbb{C}^m$ and with its $k$-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation. Reference | Related Articles | Metrics

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FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY Ang FU, Mingjin LI, Di YANG Acta mathematica scientia,Series B    2024, 44 (2): 405-419.   DOI: 10.1007/s10473-024-0201-4 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （133）      PDF       Save For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the $k$-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of $k$-point functions for the even GUE (Gaussian Unitary Ensemble) correlators. Reference | Related Articles | Metrics

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RELAXED INERTIAL METHODS FOR SOLVING SPLIT VARIATIONAL INEQUALITY PROBLEMS WITHOUT PRODUCT SPACE FORMULATION Grace Nnennaya OGWO, Chinedu IZUCHUKWU, Oluwatosin Temitope MEWOMO Acta mathematica scientia,Series B    2022, 42 (5): 1701-1733.   DOI: 10.1007/s10473-022-0501-5 Abstract （128）            Save Many methods have been proposed in the literature for solving the split variational inequality problem. Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space, or that the underlying operators are co-coercive. However, it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem. On the other hand, the co-coercive assumption of the underlying operators would preclude the potential applications of these methods. To avoid these setbacks, we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation, and for which the underlying operators are freed from the restrictive co-coercive assumption. The methods proposed, involve projections onto half-spaces only, and originate from an explicit discretization of a dynamical system, which combines both the inertial and relaxation techniques in order to achieve high convergence speed. Moreover, the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces. Furthermore, numerical implementations and comparisons are given to support our theoretical findings. Reference | Related Articles | Metrics

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ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES Jian CHEN Acta mathematica scientia,Series B    2024, 44 (2): 431-444.   DOI: 10.1007/s10473-024-0203-2 Accepted: 16 October 2023 Abstract （128）      PDF       Save In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$, where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5]. Reference | Related Articles | Metrics

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THE WEIGHTED KATO SQUARE ROOT PROBLEM OF ELLIPTIC OPERATORS HAVING A BMO ANTI-SYMMETRIC PART Wenxian MA, Sibei YANG Acta mathematica scientia,Series B    2024, 44 (2): 532-550.   DOI: 10.1007/s10473-024-0209-9 Abstract （127）      PDF       Save Let $n\ge2$ and let $L$ be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a $\mathrm{BMO}$ anti-symmetric part in $\mathbb{R}^n$. In this article, we consider the weighted Kato square root problem for $L$. More precisely, we prove that the square root $L^{1/2}$ satisfies the weighted $L^p$ estimates $\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}\le C\|\nabla f\|_{L^p_\omega (\mathbb{R}^n;\mathbb{R}^n)}$ for any $p\in(1,\infty)$ and $\omega\in A_p{(\mathbb{R}^n)}$ (the class of Muckenhoupt weights), and that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)}\le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,2+\varepsilon)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{2+\varepsilon}{p})'}(\mathbb{R}^n)$ (the class of reverse Hölder weights), where $\varepsilon\in(0,\infty)$ is a constant depending only on $n$ and the operator $L$, and where $(\frac{2+\varepsilon}{p})'$ denotes the Hölder conjugate exponent of $\frac{2+\varepsilon}{p}$. Moreover, for any given $q\in(2,\infty)$, we give a sufficient condition to obtain that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)} \le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,q)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{q}{p})'}(\mathbb{R}^n)$. As an application, we prove that when the coefficient matrix $A$ that appears in $L$ satisfies the small $\mathrm{BMO}$ condition, the Riesz transform $\nabla L^{-1/2}$ is bounded on $L^p_\omega(\mathbb{R}^n)$ for any given $p\in(1,\infty)$ and $\omega\in A_p(\mathbb{R}^n)$. Furthermore, applications to the weighted $L^2$-regularity problem with the Dirichlet or the Neumann boundary condition are also given. Reference | Related Articles | Metrics

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A SUPERLINEARLY CONVERGENT SPLITTING FEASIBLE SEQUENTIAL QUADRATIC OPTIMIZATION METHOD FOR TWO-BLOCK LARGE-SCALE SMOOTH OPTIMIZATION* Jinbao Jian, Chen Zhang, Pengjie Liu Acta mathematica scientia,Series B    2023, 43 (1): 1-24.   DOI: 10.1007/s10473-023-0101-z Abstract （125）      PDF       Save This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints. Based on the ideas of splitting and sequential quadratic optimization (SQO), a new feasible descent method for the discussed problem is proposed. First, we consider the problem of quadratic optimal (QO) approximation associated with the current feasible iteration point, and we split the QO into two small-scale QOs which can be solved in parallel. Second, a feasible descent direction for the problem is obtained and a new SQO-type method is proposed, namely, splitting feasible SQO (SF-SQO) method. Moreover, under suitable conditions, we analyse the global convergence, strong convergence and rate of superlinear convergence of the SF-SQO method. Finally, preliminary numerical experiments regarding the economic dispatch of a power system are carried out, and these show that the SF-SQO method is promising. Reference | Related Articles | Metrics

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THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE Ke JIN, Ying SHI, Huafei XIE Acta mathematica scientia,Series B    2024, 44 (2): 583-608.   DOI: 10.1007/s10473-024-0212-1 Abstract （123）      PDF       Save In this paper, we consider the semilinear elliptic equation systems $ \left\{\begin{array}{ll} -\Delta u+u=\alpha Q_{n}(x)|u|^{\alpha-2}|v|^{\beta}u &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N},\\ -\Delta v+v=\beta Q_{n}(x)|u|^{\alpha}|v|^{\beta-2}v &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N}, \end{array} \right. $ where $N\geqslant 3$, $\alpha$, $\beta>1$, $\alpha+\beta<2^{*}$, $2^{*}=\frac{2N}{N-2}$ and $Q_{n}$ are bounded given functions whose self-focusing cores $\{x\in\mathbb{R}^N|Q_n(x)>0\}$ shrink to a set with finitely many points as $n\rightarrow\infty$. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems. Reference | Related Articles | Metrics

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THE SINGULAR LIMIT OF SECOND-GRADE FLUID EQUATIONS IN A 2D EXTERIOR DOMAIN* Xiaoguang You, Aibin Zang Acta mathematica scientia,Series B    2023, 43 (3): 1333-1346.   DOI: 10.1007/s10473-023-0319-9 Abstract （122）      PDF       Save In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a well-known non-Newtonian fluid model, with two parameters: $\alpha$, which represents the length-scale, while $\nu > 0$ corresponds to the viscosity. We prove that, as $\nu, \alpha$ tend to zero, the solution of the second-grade fluid equations with suitable initial data converges to the one of Euler equations, provided that $\nu = {o}(\alpha^\frac{4}{3})$. Moreover, the convergent rate is obtained. Reference | Related Articles | Metrics

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THE GROWTH OF SOLUTIONS TO HIGHER ORDER DIFFERENTIAL EQUATIONS WITH EXPONENTIAL POLYNOMIALS AS ITS COEFFICIENTS* Zhibo Huang, Minwei Luo, Zongxuan Chen Acta mathematica scientia,Series B    2023, 43 (1): 439-449.   DOI: 10.1007/s10473-023-0124-5 Abstract （122）      PDF       Save By looking at the situation when the coefficients $P_{j}(z)$ $(j=1,2,\cdots,n-1)$ (or most of them) are exponential polynomials, we investigate the fact that all nontrivial solutions to higher order differential equations $f^{(n)}+P_{n-1}(z)f^{(n-1)}+\cdots+P_{0}(z)f=0$ are of infinite order. An exponential polynomial coefficient plays a key role in these results. Reference | Related Articles | Metrics

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EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS* Yuanyuan Luo, Dongmei Gao, Jun Wang Acta mathematica scientia,Series B    2023, 43 (1): 303-323.   DOI: 10.1007/s10473-023-0117-4 Abstract （120）      PDF       Save We study the Choquard equation $\begin{equation*}\label{a-1} -\Delta u+V(x)u=b(x)\int_{{\mathbb{R}^{3}}}{\frac{{{\left| u(y) \right|}^{2}}}{{{\left| x-y \right|}}}{\rm d}y}{u},\ x\in\mathbb{R}^{3}, \end{equation*}$ where $ V(x)=V_1(x)$, $ b(x)=b_1(x) $ for $ x_1>0 $ and $ V(x)=V_2(x), b(x)=b_2(x) $ for $ x_1<0 $, and $ V_1 $, $ V_2 $, $ b_1 $ and $ b_2 $ are periodic in each coordinate direction. Under some suitable assumptions, we prove the existence of a ground state solution of the equation. Additionally, we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation. Reference | Related Articles | Metrics

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HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS* Qingye Zhang, Chungen Liu Acta mathematica scientia,Series B    2023, 43 (3): 1195-1210.   DOI: 10.1007/s10473-023-0312-3 Abstract （119）      PDF       Save In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems $\dot{z}=JH_z(t,z)$, where the Hamiltonian function $H$ possesses the form $H(t,z)=\frac{1}{2}L(t)z\cdot z +G(t,z)$, and $G(t,z)$ is only locally defined near the origin with respect to $z$. Under some mild conditions on $L$ and $G$, we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadraticity of $G$ near the origin with respect to $z$. Reference | Related Articles | Metrics

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STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION Mohsan Raza, Hadiqa Zahid, Jinlin Liu Acta mathematica scientia,Series B    2024, 44 (4): 1244-1270.   DOI: 10.1007/s10473-024-0404-8 Abstract （119）            Save Let $q_{\lambda }\left( z\right) =1+\lambda \sinh (\zeta ),\ 0<\lambda <1/\sinh \left( 1\right) $ be a non-vanishing analytic function in the open unit disk. We introduce a subclass $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ of starlike functions which contains the functions $\mathfrak{f}$ such that $z\mathfrak{f}^{\prime }/\mathfrak{f}$ is subordinated by $q_{\lambda }$. We establish inclusion and radii results for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $. We also find a sharp bound for the third Hankel determinant for the case $\lambda =1/2$. Reference | Related Articles | Metrics