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Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality where the constant 
$$\begin{eqnarray}\min _{0\leq t\leq 1}\int _{\mathbb{R}}f(x)f(x+t)\,dx\leq 0.411||f||_{L^{1}}^{2}\quad \text{for}~f\in L^{1}(\mathbf{R}),\end{eqnarray}$$
$0.411$ cannot be replaced by 
$0.37$. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics. We show that for 
$f$ to be extremal for this inequality, we must have Our central technique for deriving this result is local perturbation of 
$$\begin{eqnarray}\max _{x_{1}\in \mathbb{R}}\min _{0\leq t\leq 1}\left[f(x_{1}-t)+f(x_{1}+t)\right]\leq \min _{x_{2}\in \mathbb{R}}\max _{0\leq t\leq 1}\left[f(x_{2}-t)+f(x_{2}+t)\right].\end{eqnarray}$$
$f$ to increase the value of the autocorrelation, while leaving 
$||f||_{L^{1}}$ unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let 
$d,n\in \mathbb{Z}^{+}$, 
$f\in L^{1}$, 
$A$ be a 
$d\times n$ matrix with real entries and columns 
$a_{i}$ for 
$1\leq i\leq n$ and 
$C$ be a constant. For a broad class of matrices 
$A$, we prove necessary conditions for 
$f$ to extremise autocorrelation inequalities of the form 
$$\begin{eqnarray}\min _{\mathbf{t}\in [0,1]^{d}}\int _{\mathbb{R}}\mathop{\prod }_{i=1}^{n}~f(x+\mathbf{t}\cdot a_{i})\,dx\leq C||f||_{L^{1}}^{n}.\end{eqnarray}$$
Let ρ be a monotone quasinorm defined on 
${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). Let T be a monotone quasilinear operator on 
${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone of λ-quasiconcave functions
where 
$$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$
$1\les p\les \infty $ and v is a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u, v, w) for which the three weighted Hardy-type inequality
holds for all λ-quasiconcave functions and all 0 < p, q ⩽ ∞.
$$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$
