Prove that the following functions are self-concordant. In your proof, restrict the function to a line, and apply the composition with logarithm rule.
((a)) f(x,y)=−log(y2−xTx) on {(x,y)∣∥x∥2<y}.
((b)) f(x,y)=−2logy−log(y2/p−x2), with p≥1, on {(x,y)∈R2∣∣x∣p<y}.
((c)) f(x,y)=−logy−log(logy−x) on {(x,y)∣ex<y}.