Averages of double shifted convolution sum of half-integral weight cusp forms

In this article, we obtain an estimate of the following sum associated to the Fourier coefficients of cusp forms of half-integral weight: 1/H Sigma(V)(h <= H) (h/H) Sigma(n <= N) lambda(f1) (n) lambda(f2) (n+h)lambda(f3) (n+2h)W (n/N) where lambda(fi) (n) are the normalized Fourier coefficients of half-integral weight cusp forms (newforms) f(i) of weight k(i) + 1/2 for i = 1,2,3 on the congruence subgroup of level 4M, k(i) >= 2 and V andW are smooth bump functions supported on [1,2]. We obtain a non-trivial upper bound estimate using a variant of circle method.