Let denote the standard unit disk in . The famous Moser–Trudinger inequality says that

holds. There is another important inequality in analysis, the Hardy inequality which claims that

holds. The one is usuall called the Hardy functional. One can immediately see that

for any . Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant such that the following

where is the unit ball in , and is the complement of with respect to the following norm .

Let us go back to the case . They then defined

where is a regular, bounded and convex domain sitting in . They then conjectured that the following

still holds for some constant where denotes the completion of with the corresponding norm associated with . Apparently, the conjecture holds true for .

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