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In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let  A  1 , …,  A   n   be bounded linear operators on a complex separable Hilbert spaceH    $\mathcal{H}$    and let  α  1 , …,  α   n   be positive real numbers such that∑   j  =  1    nα   j    =  1   $\sum\limits^{n}_{j=1}\alpha_{j}=1$   . We show that for every unitarily invariant norm,   If  f  is a non-negative function on [0, ∞) such that  f (0) = 0 andg  (  t  )  =  f  (   t   )   $g(t)=f(\sqrt{t})$    is convex, then|    |    |    ∑   j  =  1    nα   j    f  (   |    A   j|   )   |    |    |≥   |    |    |    ∑   j  ,  k  ∈   S   ℓ(   f   (α   jα   k4   α   ℓ    (  1  −   α   ℓ    )|    A   j    +   A   k    −  2   ∑   j  =  1    nα   jA   j|    )+  f   (α   jα   k    (  2   α   ℓ    −  1  )    4   α   ℓ    (  1  −   α   ℓ    )|    A   j    −   A   k|    )    )   +  f   (    |    ∑   j  =  1    nα   jA   j|    )    |    |    |$$\begin{align*}\bigg|\bigg|\bigg|\sum\limits^{n}_{j=1}\alpha_{j}f(|A_{j}|)\bigg|\bigg|\bigg|&amp;\geq\bigg|\bigg|\bigg|\sum\limits_{j,k\in S_{\ell}}\bigg(f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}}{4\alpha_{\ell}(1-\alpha_{\ell})}}\;\bigg|A_{j}+A_{k}-2\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\\&amp;\qquad+f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}(2\alpha_{\ell}-1)}{4\alpha_{\ell}(1-\alpha_{\ell})}}|A_{j}-A_{k}|\bigg)\bigg)+f\bigg(\bigg|\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\bigg|\bigg|\bigg|\end{align*}$$for ℓ = 1, …,  n .    If  f  is a non-negative function on [0, ∞) such thatg  (  t  )  =  f  (   t   )   $g(t)=f(\sqrt{t})$    is concave, then the inverse inequality holds. Here, the symbol  S  ℓ  = {1, …,  n } ∖ {ℓ} for ℓ ∈ {1, …,  n }.

Clarkson inequalities related to convex and concave functions