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I want to find upper and lower bounds for $W(\frac{ln(x)}{a})$, where $a$ is a positive constant. Is there any approximation or upper and lower bounds for this function using only elementary functions? Bounds don't need to be tight and even loose bounds will be helpful.

I came across this in my search for a solution where it states that if $x > e$ then $$ \log x - \log \log x < W(x) < \log x $$ However, I couldn't find any approximation for the case when $ e \geq x > 0$.

P.S. a trivial lower bound for the range $ e \geq x > 0$ is $\frac {x}{e}$ but I suspect there might be a tighter lower bound.

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Let $x>0$, by the definition and the monotonicity of $W$, $$ x = W(xe^x ) > W(x + x^2 ). $$ This gives $$ W(x) < \frac{{\sqrt {4x + 1} - 1}}{2} $$ for all $x>0$. From this, we also have $$ W(x) = \log \left( {\frac{x}{{W(x)}}} \right) > \log \left( {\frac{{2x}}{{\sqrt {4x + 1} - 1}}} \right) = \log \left( \frac{{\sqrt {4x + 1} + 1}}{2}\right) $$ for all $x>0$.

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