The purpose of this paper is to introduce and discuss the concept of topical functions on upward sets. We give characterizations of topical functions in terms of upward sets.

In this note, we prove that the Toeplitz-type Operator $\Theta_\alpha^b$ generated by the generalized fractional integral, Calderón-Zygmund operator and VMO function is bounded from $L^{p,\lambda} (R^n)$ to $L^{q,\mu} (R^n)$. We also show that under some conditions $\Theta_\alpha^b f \in  V L^{q,\mu} (B_R)$, the vanishing-Morrey space.

Let $A$, $B$ be two unital $C^*$−algebras. By using fixed point methods, we prove that every almost unital almost linear mapping $h : A \to B$ which satisfies $h(2^nuy)= h(2^nu)h(y)$ for all $u \in U(A)$, all $y \in A$, and all $n=0,1,2, \cdots$, is a homomorphism. Also, we establish the generalized Hyers–Ulam–Rassias stability of $*$−homomorphisms on unital $C^*$−algebras.

Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space $\sum_2$ containing transitive points in a simpler way with the help of a different metric. We also provide two examples, which support this new definition.

If $P(z)$ is a polynomial of degree $n$ which does not vanish in $|z|<1$, then it is recently proved by Rather [Jour. Ineq. Pure and Appl. Math., 9 (2008), Issue 4, Art. 103] that for every $\gamma < 0$ and every real or complex number $\alpha$ with $|\alpha| \geq 1,$$$\Big\{\int_0^{2\pi}|D_\alpha P(e^{i\theta})|^\gamma d\theta\Big\}^{1/\gamma}\leq n(|\alpha|+1) C_\gamma\Big\{\int_0^{2\pi}|P(e^{i\theta})|^\gamma d\theta\Big\}^{1/\gamma},$$$$C_\gamma=\Big\{\frac{1}{2\pi}\int_0^{2\pi}|1+e^{i\beta}|^\gamma d\beta\Big\}^{-1/\gamma},$$where $D_\alpha P(z)$ denotes the polar derivative of $P(z)$ with respect to $\alpha$. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 (1988), 306-313] as a special case.

We consider complex-valued functions $f \in  L^1(\mathbf{R}^2_+)$, where $\mathbf{R}_+ := [0,\infty)$, and prove sufficient conditions under which the double sine Fourier transform $\hat{f}_{ss}$ and the double cosine Fourier transform $\hat{f}_{cc}$  belong to one of the two-dimensional Lipschitz classes $Lip(\alpha,\beta )$ for some $0 < \alpha,\beta \leq 1$; or to one of the Zygmund classes Zyg$(\alpha,\beta )$ for some $0 < \alpha,\beta  \leq 2$. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions $f \in L^1(\mathbf{R}^2_+)$.

In this paper, the authors study the boundedness of the operator $\mu^b_\Omega$, the commutator generated by a function $b \in \text{Lip}_\beta (\mathbf{R}^n)$ $(0<\beta<1)$ and the Marcinkiewicz integral $\mu_\Omega$ on weighted Herz-type Hardy spaces.

Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].

A theoretical framework for the calculation of Hausdorff measure of self-similar sets satisfying OSC has been established.