Optimality of constants in power-weighted Birman-Hardy-Rellich-Type inequalities with logarithmic refinements

The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, alpha) and B(m, alpha), m is an element of N, alpha is an element of R, of the form A(m, alpha) = 4(-m) Pi(m)(j = 1) (2j - 1 - alpha)(2), B(m, alpha) = 4(-m) Sigma(m)(k = 1) Pi(m)(j = 1j not equal k)(2j - 1 - alpha)(2), in the power-weighted Birman-Hardy-Rellich-type integral inequalities with logarithmic refinement terms recently proved in [41], namely, integral(rho)(0) dx x(proportional to) vertical bar f((m) )vertical bar(2) >= A(m, alpha) integral(rho)(0) dx x(proportional to-2m)vertical bar f(x)vertical bar(2) + B(m, alpha) Sigma(N)(k=1) integral(rho)(0) dx x(proportional to-2m) Pi(k)(p=1) [ln(p)(gamma/x)](-2)vertical bar f(x)vertical bar(2), f is an element of C-0(infinity) ((0, rho)), m, N is an element of N, alpha is an element of R, rho, gamma is an element of(0, infinity), gamma >= e N rho. Here the iterated logarithms are given by ln(1)(.) = ln(.), ln(j+1)(.) = ln(ln(j)(.)), and the iterated exponentials are defined via e(0) = 0, e(j+1 )= e(ej), j is an element of N-0 = N boolean OR {0}. Moreover, we prove the analogous sequence of inequalities on the exterior interval (r, infinity) for f is an element of C-0(infinity)((r, infinity)), r is an element of (0, infinity), and once again prove optimality of the constants involved.