For a positive integer $N$, let $X_0^*(N)$ denote the quotient curve of $X_0(N)$ by the Atkin-Lehmer involutions. In this paper, we determine the trigonality of $X_0^*(N)$ for all $N$. It turns out that there are seven values of $N$ for which $X_0^*(N)$ is a non-trivial trigonal curve.

We give elementary proofs of the results given by Ballet, Faith-Herbera, Goblot, Vámos, Warner, on linearly compact domains.

We give a simple criterion for the solvability in integers of the negative Pell equation $x^2 - dy^2 = -1$ by means of primitive Pythagorean triples. The method of proof allows us to also solve the problem of determinig the fundamental units of negative norm in real quadratic fields.

A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional $|a_3 - \mu a_2^2|$ for certain classes of univalent functions. In this paper, we consider the class $\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions which is defined by using subordination for analytic functions $\varphi$ and $\psi$ on the unit disc. Our main object is to provide bounds of the quantity $a_3 - \mu a_2^2$ for functions $f(z) = z + a_2 z^2 + a_3 z^3 + \dotsb$ in $\mathcal{C}(\varphi, \psi)$ in terms of $\varphi$ and $\psi$, where $\mu$ is a real constant. We also show that the class $\mathcal{C}(\varphi, \psi)$ is closed under the convolution operation by convex functions, or starlike functions of order $1/2$ when $\varphi$ and $\psi$ satisfy some mild conditions.