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In this article, we investigate the first sign change of the sequence $\\{a_f(p^{\\alpha})a_g(p^{\\alpha}) \\}_{p^{\\alpha} \\in \\N, \\alpha \\le 2}$, where $p$ is a prime number. We further study the non-vanishing of the sequence $\\{a_f(n)a_g(n) \\}_{n \\in \\N}$ and derive bounds for first non-vanishing term in this sequence. 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Also let $a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients of $f$ and $g$ respectively. In this article, we investigate the first sign change of the sequence $\\{a_f(p^{\\alpha})a_g(p^{\\alpha}) \\}_{p^{\\alpha} \\in \\N, \\alpha \\le 2}$, where $p$ is a prime number. We further study the non-vanishing of the sequence $\\{a_f(n)a_g(n) \\}_{n \\in \\N}$ and derive bounds for first non-vanishing term in this sequence. 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