论文标题
涉及除数函数的泰勒级数的不平等
Inequalities for Taylor series involving the divisor function
论文作者
论文摘要
令$$ t（q）= \ sum_ {k = 1}^\ infty d（k）q^k，\ quad | q | <1，$$，其中$ d（k）$表示自然数$ k $的正数差异。我们介绍按$ t $定义的功能的单调性属性。 更具体地说，我们证明了$$ h（q）：= t（q） - \ frac {\ log（1-q）} {\ log（q）} $$ $（0,1）$严格增加，而$$ f（q）：= \ frac {1-q} {q} {q} \，h（q）$$ 严格减少$（0,1）$。 然后将这些结果应用于获得各种不等式，其中之一指出双重质量$$α\，\ frac {q} {1-q} {1-q}+\ frac {\ log（1-q）} {\ log log（q）} <t（q）<t（q） β\，\ frac {q} {1-q}+\ frac {\ log（1-q）} {\ log（q）}，\ quad 0 <q <1，$$具有最佳常数因子$α=γ$和$β= 1 $。在这里，$γ$表示Euler的常数。这是塞勒姆的结果，他证明了$α= 1/2 $和$β= 1 $的不平等现象。
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ 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 proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ α\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< β\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}, \quad 0<q<1, $$ holds with the best possible constant factors $α=γ$ and $β=1$. Here, $γ$ denotes Euler's constant. This refines a result of Salem, who proved the inequalities with $α=1/2$ and $β=1$.
