Search Results (343)

This paper presents a new subclass ${\mathcal{BS}}_{k-1}^{*}$ of starlike functions with respect to symmetric points associated with a domain bounded by an epicycloid with $k-1$ cusps. First, we derive bounds for the first five Taylor coefficients of functions in this subclass and obtain the first three coefficients of the corresponding logarithmic function. Moreover, for some special values, upper and lower bounds are established for several important functionals and determinants associated with the coefficients of functions in this subclass. Full article

Chain topology offers a chemistry-preserving route to tune block copolymer (BCP) self-assembly by modifying intrachain correlations and relaxation pathways without changing monomer interactions. Here, we perform a unit-matched comparison of four lamella-forming AB architectures reconstructed from an identical constitutive diblock unit ($N_0$): a linear diblock (DB), a linear multiblock (MB), a comb-like architecture (CB), and a star-like architecture (SB). Using dynamical density functional theory (DDFT), we quantify topology-dependent bulk ordering thresholds and show that architectural reconfiguration systematically stabilizes the ordered phase, reducing the order–disorder transition relative to DB (MB/CB/SB $\approx 0.793/0.762/0.752$ of the diblock value), in semi-quantitative agreement with random phase approximation (RPA) spinodal trends. We also compare topology-dependent directed self-assembly in a common trench geometry under matched reduced quench depth $\Delta \left(\chi N_0\right)=\chi N_0-{\left(\chi N_0\right)}_{\mathrm{ODT}}$, thereby isolating kinetic differences at comparable thermodynamic distance from bulk ordering. A Fourier-based alignment order parameter $\alpha \left(t\right)$ reveals sigmoidal alignment kinetics over decades in time and is well captured by a logistic form in $lnt$, enabling compact descriptors ($t_{50}$, $t_{90}$, and a steepness parameter k) that separate alignment onset from late-stage defect annihilation, while selective sidewalls robustly template sidewall-parallel lamellae across all topologies, the late-stage kinetics remain strongly connectivity dependent and can exhibit long-tailed completion associated with slow late-stage defect annihilation. These results demonstrate a dual role of topology in DSA: lowering the segregation strength required for bulk ordering while reshaping defect-mediated alignment pathways under confinement. Full article

In this paper, we study a generalization of the Graham–Kohr extension operator, ${\mathsf{\Psi }}_{n,\alpha ,\beta }^{\gamma }\left(f\right)$, which maps functions defined on the unit disk into holomorphic mappings in the unit ball ${\mathbb{B}}^n$. Using the theory of Loewner chains, we show that, under suitable conditions, this operator can be embedded as the first element of a Loewner chain while preserving geometric properties. In addition, for suitable choices of the parameters, we establish subordination relations among starlike functions. Full article

In this work, we provide a convolution type operator ${\mathsf{\Lambda }}_{\nu ,b}$ that is produced by the generalized Marcum Q-function and examine how it maps to various Janowski-type subclasses of harmonic univalent functions. Since the Marcum Q-function has an integral form via the lower incomplete gamma function, the convolution operator ${\mathsf{\Lambda }}_{\nu ,b}$ can be understood as a fractional type integral operator operating on the coefficients of harmonic mappings. Applying ${\mathsf{\Lambda }}_{\nu ,b}$ to harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$, we derive coefficient inequalities, and inclusion relations for various subclasses of harmonic and analytic univalent functions. In particular, we give sufficient conditions for ${\mathsf{\Lambda }}_{\nu ,b}\left(f\right)$ to belong to Janowski-starlike families such as ${\mathcal{S}}_H^{\ast }(F,G)$, ${\mathcal{K}}_H^0$, and ${\mathcal{R}}_H(F,G)$. Closure properties of the Janowski class under the proposed operator are also established. Numerical tables and examples confirm the inclusion results, and graphical plots illustrate how the operator reshapes the image domains for different parameter pairs $(\nu ,b)$. Numerical illustrations are provided to visualize the geometric steering effect induced by the Marcum Q-function and its fractional-order damping behavior. Full article

In this paper, we investigate a class of analytic functions associated with the generalized Marcum Q-function and its Alexander transform. We establish sufficient conditions under which these functions exhibit important geometric properties in the open unit disk, including strong starlikeness, strong convexity, and pre-starlikeness. The results presented are believed to be new and are supported by illustrative examples and consequences. Full article

Recall that in On a general second order derivative, Sci. Rep. Tokyo Kyoiku Daigaku A, 5(124–127)(1956), 111–114, Ozaki, Ono and Umezawa proved a result that if $f\left(z\right)$ is analytic and satisfies $|f^{″}\left(z\right)|<1$ in the unit disc $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$, then $|f^{\prime }\left(z\right)-1|<1$ and so, $f\left(z\right)$ is univalent in $\mathbb{D}$, because $\mathfrak{Re}\left\{f^{\prime }\left(z\right)\right\}>0$ in $\mathbb{D}$ implies univalence by the Noshiro–Warschawski Theorem. In this paper, we obtain another sufficient condition for univalence of $f\left(z\right)$ by applying a hypothesis for modulus of $arg\left\{f^{″}\left(z\right)\right\}$. Full article

This paper introduces coefficient functionals for a new subclass (${\mathcal{S}}_{tanh}^{*})$ of starlike functions associated with the tangent hyperbolic function, including the first four sharp coefficient bounds, the Fekete-Szegő problem, Zalcman inequalities, and Hankel determinants. For this class, logarithmic and inverse problems are also studied. Furthermore, we define families of functions that are related to the functions $1+sin\mu ,$${\left(1+\alpha \mu \right)}^2,1+{\displaystyle \frac{\mu }{1-\beta {\mu }^2}}$, represented by ${\mathcal{Y}}_{sin},{\mathcal{Y}}_{\alpha }$ and ${\mathcal{Y}}_{\beta }$, respectively. Using the Schwarz-Pick lemma and the theory of subordination, involving the function $1+{\displaystyle \frac{1}{2}}tanh\mu$, we find the majorization radii and construct majorization results of the form $\left|g^{\prime }\left(\mu \right)\right|\le \left|h^{\prime }\left(\mu \right)\right|$ for functions g majorized by h. Through graphical analysis, we also demonstrate that our defined class ${\mathcal{S}}_{tanh}^{*}$ is non-empty, which validates our study in this article. Full article

In this paper, we introduce and investigate a new class of analytic functions generated by Euler polynomials through a suitable normalization. Using classical tools from geometric function theory, including coefficient monotonicity, Fejér-type inequalities, MacGregor’s criteria, and Ozaki’s close-to-convexity condition, we establish sufficient conditions for the univalence, starlikeness, convexity, and close-to-convexity of the proposed Euler-polynomial-based normalized function. Sharp radius results for starlikeness, convexity, and close-to-convexity in the disk ${\mathbb{D}}_{1/2}$ are derived by exploiting refined coefficient bounds involving higher-order Euler polynomial terms. Several illustrative examples and graphical demonstrations are provided to verify the theoretical findings. The results obtained extend the known geometric properties of special function-based analytic classes and offer a new perspective on the geometric behavior of Euler polynomials in the unit disk. Full article

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion bounds for the mappings and their complex derivatives. The exact region where the log harmonic mappings of the form $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$, with h being starlike analytic, give the starlike image is determined. Subordination relations for logarithmic derivatives are established, connecting the mappings with the Schwarz lemma and the Carathéodory class. Furthermore, we obtain the growth estimates for the underlying starlike functions h and their derivatives, as well as accurate inequalities governing the arclength of the circle image under log-harmonic mappings. These findings contribute to the geometric function theory of log-harmonic mappings. Full article

A model of nanoparticles has been designed to partially resemble self-similar ferroelastic star-like domain textures. Numerical computations have been used to find the equilibrium configurations of magnetisation in such systems. As expected from the symmetry, the self-similar initial states give room to other types of domain structure as a function of the star parameters. When relaxed without an external field, the self-similar pattern mostly turns into a massive vortex in the centre with radially oriented domains in the star’s peripheral arms. In contrast, a random initial state ends up in a configuration of a triple valve with one input and two outputs, or vice versa, analogous to logical gates. A treatment with an in-plane magnetic field always leads to the valve configuration. The triple-valve states turn out stable, whereas the vortex ones are metastable. The results may be in the design of magnetic-based logic devices. Full article

This work introduces new bi-univalent function classes defined using the fractional q-Ruscheweyh operator and characterized by subordination to q-Hermite polynomials. We derive coefficient bounds and Fekete–Szegö inequalities for these classes and show that our results generalize several earlier findings in both the classical and q-analytic settings. The approach highlights the effectiveness of q-Hermite structures in analyzing operator-defined subclasses of bi-univalent functions. Full article

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander transform. For these functions, we establish sufficient conditions ensuring membership in the classes of lemniscate starlike and lemniscate convex functions. Special attention is given to the case $\nu =1$, where explicit admissible parameter ranges for b are derived. We further examine inclusion relations between these normalized analytic forms and lemniscate subclasses, complemented by several corollaries, illustrative examples, and graphical visualizations. These results extend and enrich the geometric function theory of special functions related to the generalized Marcum Q-function. Full article

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of these normalized functions so these functions are starlike and convex of order $\gamma$ and type $\delta$. Full article

In this paper, we introduce a new class of analytic functions, denoted by $\mathfrak{S}(\nu ,{\phi }_{\vartheta ,e})$, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica^TM codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper. Full article

For the function solution to the well-known homogeneous Bessel differential equation, we utilized a normalized form of this function to define a certain operator on a subclass of analytic functions. Using this operator, we introduced various subordination properties. We also examined the sufficient starlikeness conditions and provided some estimates for a specific subclass of univalent functions defined in the unit disc. Full article