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Suppose $u$ and $v$ are subharmonic on a bounded domain $G$ of $\mathbb{R}^{n}$ with $n\geq2$, and $w=u-v.$ If $w$ is also subharmonic and defined everywhere on $G$ (we exclude the case $u$ or $v$ is harmonic), can we say that $u$ and $v$ differ by a constant?

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I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?

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  • $\begingroup$ Thanks for noticing it. You are right! I edited and excluded this case. $\endgroup$ – M. Rahmat Dec 25 '18 at 6:15
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    $\begingroup$ @M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer. $\endgroup$ – Kavi Rama Murthy Dec 25 '18 at 6:19

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