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For a fixed $1\le p<+\infty$ denote by $\Vert\cdot\Vert_p$ the usual norm inthe space $l_p$ (or $L_p$). In this paper we prove that for all real numbers$p$ and $q$ such that $2\le p\le q$ holds  $$ 2(\Vert x\Vert_p^q+\Vert y\Vert_p^q)\le \Vert x+y\Vert_p^q +\Vertx-y\Vert_p^q  $$ for all nonnegative sequences $x=\{x_n\},y=\{y_n\}$ in $l_p$ (ornonnegative functions $x,y$ in $L_p$). Note that the above inequality with$p=q\ge 2$ reduces to the well known Clarkson's inequality. If in addition,holds $x_i\ge y_i$ for each $i=1,2,...$ (or $x\ge y$ a.e. in $L_p$), then weestablish an improvement of the above inequality.

Clarkson's type inequalities for positive $l_p$ sequences with $p\ge 2$