TITLE: Bound on an expression involving J-function coefficients
QUESTION [7 upvotes]: I would like to show that 
    $$(m+1)\,c_m- \sum_{n=1}^{m-1} c_n \,\sigma_{m-n} > 0$$
for all $m$, where $c_i$ is the $i$th coefficient of Klein's $J$-function
    $$J(q)= \frac{1728 \;E_4^3(q) }{E_4^3(q)-E_6^2(q)}-744$$
(in terms of Eisenstein series $E_4$ and $E_6$) and $\sigma_i$ is minus the $i$th coefficient of the second order Eisenstein series $E_2$. Both are positive numbers, bounded respectively by
    $$c_n \leqslant \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\;n^{3/4}}$$
$$c_n \geqslant \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\;n^{3/4}}\, K$$
$$\sigma_n < 24 \left[e^{\gamma}\,n\,\text{ln}(\text{ln}(n))+\frac{0.6483\, n}{\text{ln}(\text{ln}(n))}\right]\,\qquad (n\geqslant 3)$$ 
where $\gamma$ is Euler's constant ($\sim 0.6$) and $K\sim 0.98$. I could also use the cruder bound
    $$c_n \leqslant e^{4\pi \sqrt{n+1}}\,.$$
I've tried bounding the quotient of the two terms, Taylor-expanding and taking a limit to show that it goes to zero, but that's only valid for small values of $n$ and I'm interested in all kinds of values. Numerical evidence suggests this quotient goes to something like 1, but I don't know how to show that, if I must refrain from Taylor-expanding. (Also, replacing the sum by $(m-1)c_{m-1}\sigma_1$ is much too brutal).
Any ideas are welcome, and if there's any interesting information on the coefficients that I'm not aware of, I would appreciate also! For example, knowledge of the relations between successive coefficients might enable one to make a recursive type of argument, but I haven't found any such thing.
Edit: To be more complete and answer David Loeffler's comment, this expression appears in the modular form of weight 2
    $$q\partial_q\,J(q)+E_2\left(J(q)+24\right)=\sum_{m=1}^{\infty}F(m)\,q^m$$
with $F(m)=(m+1)\,c_m- \sum_{n=1}^{m-1} c_n \,\sigma_{m-n}-\sigma_{m+1}-\sigma_1\sigma_m$
(I'm assuming the sigma terms are negligible).

REPLY [3 votes]: Let us define $j=J+744=\frac{1728E_4^3}{E_4^3-E_6^2}$. Then, as noted by OP, the question is equivalent to positivity of coefficients of
$$
f=q\frac{d}{dq}J+E_2(J+24)=q\frac{d}{dq}j+E_2j-720E_2.
$$
Next, by this answer, we have
$$
q\frac{d}{dq}j=-\frac{E_6}{E_4}j,
$$
therefore
$$
f=j\left(E_2-\frac{E_6}{E_4}\right)-720E_2.
$$
On the other hand, by Ramanujan identities (see here, for example), we have
$$
q\frac{d}{dq}E_4=\frac{E_2E_4-E_6}{3},
$$
so
$$
f=3j\frac{q\frac{d}{dq}E_4}{E_4}-720E_2=5184\frac{E_4^2q\frac{d}{dq}E_4}{E_4^3-E_6^2}-720E_2.
$$
As coefficients of $E_4$ are all non-negative, so are coefficients of its derivative. Also,
$$
\frac{1}{E_4^3-E_6^2}=\frac{1}{\Delta}=q^{-1}\prod_{n=1}^{+\infty}(1-q^n)^{-24}
$$
has positive coefficients. Due to the fact that coefficients of $E_2$ (apart from the $q^0$ term) are non-positive, we obtain the desired inequality.