By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established.

The function 1 divided by "x"[superscript 2] minus "e"[superscript"-x"] divided by (1 minus "e"[superscript"-x"])[superscript 2] for "x" greater than 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function "t" divided by "e"[superscript "at"] minus "e"[superscript"(a-1)""t"] for "a"…

This note further discusses the functional equation and discusses two generalizations of it.

In this short note, a mathematical proposition on a functional equation for f(xy)=xf(y) + yf(x)for x,y [does not equal] 0, which is encountered in calculus, is generalized step by step. These steps involve continuity, differentiability, a functional equation, an ordinary differential linear equation of the first order, and relationships between…

For any nonnegative integer "k" and natural numbers "n" and "m," the equations presented in this paper demonstrate the inequalities obtained on the ratio for the geometric means of a positive arithmetic sequence with unit difference, where alpha epsilon [vertical bar]0,1[vertical bar] is a constant. Using the ideas and methods of Chen (2002),…

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.

In this note involving mathematical induction, an identity involving the combinatorial numbers and the partial sums of the harmonic and related series is verified. (Author)