Inequalities for Taylor series involving the divisor function

Let $$T(q) = \sum\limits_{k 1}^\infty {d(k){q^k},\,\,\,\,\left| q \right| < 1,} $$ where d(k) denotes the number of positive divisors natural k. We present monotonicity properties functions defined in terms T. More specifically, we prove that $$H(q) T(q) - {{\log (1 q)} \over {\log (q)}}$$ is strictly increasing on (0, 1), while $$F(q) {{1 q} q}H(q)$$ decreasing 1). These results are then applied to obtain various inequalities, one which states double inequality $$\alpha {q {1 q}} + (q)}} \beta (q)}},\,\,\,\,\,\,0 1,$$ holds with best possible constant factors α γ and β 1. Here, Euler’s constant. This refines a result Salem, who proved inequalities 2}$$