Let $\mathscr{A}$ be the class of analytic functions in the unit disk D with the normalization f(0) = f′(0) − 1 = 0. For λ > 0, denote by $\mathscr{M}$(λ) the class of functions f $\in$ $\mathscr{A}$ which satisfy the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq \lambda,\quad z\in \mathbf{D}.$$ We show that functions in $\mathscr{M}$(1) are univalent in D and we present one parameter family of functions in $\mathscr{M}$(1) that are also starlike in D. In addition to certain inclusion results, we also present characterization formula, necessary and sufficient coefficient conditions for functions in $\mathscr{M}$(λ), and a radius property of $\mathscr{M}$(1).

We consider a metric d_L on the Teichmüller space T(R₀) defined by the length spectrum of Riemann surfaces. H. Shiga proved that d_L defines the same topology as that of the Teichmüller metric d_T on T(R₀) if a Riemann surface R₀ can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded from above and below.

In this paper, we show that there exists a Riemann surface R₀ of infinite type such that R₀ cannot be decomposed into such pairs of pants, whereas the two metrics define the same topology on T(R₀). We also give a sufficient condition for these metrics to have different topologies on T(R₀), which is a generalization of a result given by Liu-Sun-Wei.

In this paper we present various norm inequalities in framework Sobolev spaces by using Hölder's inequality. In particular, we imply some of the corresponding results of Saitoh whose proofs were based on Aronszajn's theory of reproducing kernels.

We classify Hopf hypersurfaces of non-flat complex space forms CP^m(4) and CH^m(-4), denoted jointly by CQ^m(4c), that are of 2-type in the sense of B. Y. Chen, via the embedding into a suitable (pseudo) Euclidean space of Hermitian matrices by projection operators. This complements and extends earlier classifications by Martinez and Ros (the minimal case) and Udagawa (the CMC case), who studied only hypersurfaces of CP^m and assumed them to have constant mean curvature instead of being Hopf. Moreover, we rectify some claims in Udagawa's paper to give a complete classification of constant-mean-curvature-hypersurfaces of 2-type. We also derive a certain characterization of CMC Hopf hypersurfaces which are of 3-type and mass-symmetric in a naturally-defined hyperquadric containing the image of CQ^m(4c) via these embeddings. The classification of such hypersurfaces is done in CQ²(4c), under an additional assumption in the hyperbolic case that the mean curvature is not equal to ±2/3. In the process we show that every standard example of class B in CQ^m(4c) is mass-symmetric and we determine its Chen-type.

In this paper, we investigate properties of finite order transcendental meromorphic solutions and rational solutions of difference Painlevé I and II equations.

We study trajectories for Sasakian magnetic fields which are also circles of positive geodesic curvature on geodesic spheres in a complex projective space. Investigating their extrinsic shapes we give a condition for them to be closed. By use of information on lengths of circles on a complex projective space, we give their lengths, and estimate the bottom of the length spectrum of circular trajectories.

An F-stationary map is a critical point of the F-energy with respect to variations in the domain. It is a generalization of F-harmonic maps. In [11, 10], we discuss the theorems of Liouville type and the monotonicity of F-stationary maps. In this paper, we discuss the stabilities of F-stationary maps from submanifolds of spheres and the Euclidean spaces.

We compute the homotopy groups of the space of self maps of the 3 dimensional projective space.

We investigate parallel isotropic immersions of an open submanifold of either the Cayley projective plane CayP²(α) or its noncompact dual into a real space form $\tilde{M}^n(\tilde{c})$, and give a characterization of the first standard embedding of CayP²(α) into $\tilde{M}^n(\tilde{c})$ in terms of a particular class of Frenet curves of order 2.