We will consider some inequalities on operator means for more than three operators, for instance, ALM and BMP geometric means will be considered. Moreover, log-Euclidean and logarithmic means for several operators will be treated.

We give a simultaneous extension of Diaz-Metcalf and Buzano inequalities: Let $z_1,\ldots,z_m$ be nonzero vectors in a Hilbert space $\mathscr{H}$. Suppose that $x_1,\ldots,x_n \in \mathscr{H}$ satisfy that for each $j=1,\ldots,m$ there exists a constant $r_j$ such that $0 \le r_j \le \frac{\mathop{\mathrm{Re}}{\left\langle{x_i},{z_j}\right\rangle}}{\left\|{x_i}\right\|}$ for $i=1,\ldots,n$. If $y_1,y_2 \in \mathscr{H}$ satisfy ${\left\langle{y_k},{z_j}\right\rangle}=0$ for $k=1,2$ and $j=1,\ldots,m$, then $${\left|{\left\langle{\sum x_i},{y_1}\right\rangle} {\left\langle{\sum x_i},{y_2}\right\rangle}\right|} + \left(\sum \frac{r_j^2}{c_j}\right) \left(\sum {\left\|{x_i}\right\|}\right)^2 \mathcal{B}\left({y_1}{y_2}\right) \le \mathcal{B}\left({y_1}{y_2}\right) \left\|{\sum {x_i}}\right\|^2,$$ where $\mathcal{B}\left({y_1},{y_2}\right) :=\frac12(\left\|{y_1}\right\| \left\|{y_2}\right\| +{\left|{\left\langle{y_1},{y_2}\right\rangle}\right|})$ and $c_j = \sum_h|{\left\langle{z_h},{z_j}\right\rangle}|$ for $j=1, \ldots, m$.

In this paper, we present new Garkavi-Klee type characterizations of inner product spaces using the notion of generalized centers of three points sets introduced by using absolute normalized norms.

We consider isosceles orthogonality and Birkhoff orthogonality, which are the most used notions of generalized orthogonality. In 2006, Ji and Wu introduced a geometric constant $D(X)$ to give a quantitative characterization of the difference between these two orthogonality types. From their results, we have that $D(X)=D(X^*)$ holds for any symmetric Minkowski plane. On the other hand, for the James constant $J(X)$, Saito, Sato and Tanaka recently showed that if the norm of a two-dimensional space $X$ is absolute and symmetric then $J(X)=J(X^*)$ holds. In this paper, we consider the constant $D(X,\lambda)$ such that $D(X)=\inf_{\lambda \in \mathbb{R}}D(X,\lambda)$ and obtain that in the same situation $D(X,\lambda)=D(X^*,\lambda)$ holds for any $\lambda \in (0,1)$.

In this paper, we show several operator inequalities involving the Hadamard product and the Ando-Li-Mathias mean of $n$ positive operators on a Hilbert space, which are regarded as $n$-variable versions of the Fiedler-Ando theorem. As an application, we show an $n$-variable version of Fiedler type inequality via the Ando-Li-Mathias mean.

In this paper, we consider the split common fixed point problem with families of mappings in Banach spaces. Then using the hybrid method and the shrinking projection method, we prove strong convergence theorems for finding a solution of the split common null point problem with families of mappings in Banach spaces.

The concept of widely more generalized hybrid mapping introduced by Kawasaki and Takahashi in [6] in the case of Hilbert spaces. We also use this definition in the case of Banach spaces. In this paper we discuss some strong convergence theorems of orbits to a fixed point for widely more generalized hybrid mappings.

We give a confirmation that the Möbius ball of any real inner product space is a gyrovector space by using only elementary hand calculation. Some remarks to [2] will also be made.

A refinement of the Löwner--Heinz inequality has been discussed by Moslehian--Najafi. In the preceding paper, we improved it and extended to the Furuta inequality. In this note, we give a further extension for the grand Furuta inequality. We also discuss it for operator means. A refinement of the arithmetic-geometric mean inequality is obtained.

Recently, Pečarić and Rajić established a generalization of the Dunkl-Williams inequality for $n$ elements in a Banach space. In this note we show a refinement of this inequality.

We consider the pointwise multipliers on Musielak-Orlicz spaces. We treat a wide class of Musielak-Orlicz spaces with generalized Young functions which include quasi-normed spaces.

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space. We shall characterize the maximal topological linear subspace $M_{\infty}$ of $L_0(\Omega, \mathcal{A}, \mu)$ in the case where $\mu(\Omega)=+\infty$. $M_{\infty}$ is the truncated $L_{\infty}$ space which is open and closed in $L_0(\Omega, \mathcal{A}, \mu)$. In the case where $\Omega=\textbf{N}$(natural numbers), $\mu(A)=\sharp A=$ the cardinal number of $A$, the maximal linear subspace of $L_0(\textbf{N}, \mu)$ is $\ell_{\infty}$.

In this note we compute the boundary of some generalized numerical range $W_q(A)$ of a $4 \times 4$ Toeplitz nilpotent matrix $A$. We also provide a program to plot $W_q(A)$ by using ``Mathematica".

We study the viscosity approximation method for a sequence of quasinonexpansive mappings with contraction-like mappings. We establish a strong convergence theorem and then we apply our result to approximate a solution of a split feasibility problem and a fixed point of a Lipschitz continuous pseudo-contraction.