Relaxation phenomenon, thermodynamics, and phase transformation of water in nanopores has been studied by differential scanning calorimetry, usually on heating a precooled sample. Interpretation of such results is ambiguous, because the data do not indicate whether or not there is a thermal hysteresis between the heating and cooling paths. We argue that measurements on both the cooling and heating paths are needed, particularly for complex systems, and also measurements of the complex quantity are needed to ascertain that the Kramers-Kronig relations for a relaxation process are obeyed. We report a study of the real and imaginary components of the complex specific heat, C(p)' and C(p)", and the time-dependent C(p,app) of water confined to 2.4 nm diameter cylindrical pores on both the cooling and heating paths, and use different thermal histories. C(p,app) of nanoconfined water shows two exothermic peaks during cooling below 255 K at 12 K/h and only one endothermic peak on heating, and the enthalpy change determined from the exotherm is more than that determined from the endotherm. C(p,app) and C(p)' of the partially crystallized water is higher at 240 K than at 275 K on the cooling path, and C(p,app) and C(p)' of the partially crystalline water are lower at 240 K than at 275 K on the heating path, thus showing a thermal hysteresis in this range. Studies by using 60 K/h cooling and heating rates and the effect of heat treatment at selected temperatures confirm that the features observed are due to slow crystallization and slow melting. The endotherm observed on the heating path with onset at 220 K and peak at 227 K is due to gradual melting of the ice in nanopores, and not due to glass-softening transition, a liquid-liquid transition, or an approach toward the conjectured critical point of the supercooled water in the 2.4 nm pores.