Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who proved the inequalities...

We say that the function $f:[a,b]\to ℝ$ is under the chord if $$\frac{\left(b-t\right)f\left(a\right)+\left(t-a\right)f\left(b\right)}{b-a}\ge f\left(t\right)$$ for any $t\in [a,b]$. In this paper we proved amongst other that $${\int }_a^{b}u\left(t\right)df\left(t\right)\ge \frac{f\left(b\right)-f\left(a\right)}{b-a}{\int }_a^{b}u\left(t\right)dt$$ provided that $u:[a,b]\to ℝ$ is monotonic nondecreasing and $f:[a,b]\to ℝ$ is continuous and under the chord. Some particular cases for the weighted integrals in connection with the Fejér inequalities are provided. Applications for continuous functions of selfadjoint operators on Hilbert spaces are also given.

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $\alpha \le {\sum }_{j=1}^{n-1}1/(n²-j²)sin\left(jx\right)\le \beta$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0<{\sum }_{j=1}^{n-1}(n²-j²)sin\left(jx\right)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].