frequency translation or pitch shifting through linear interpolation in frequency domain
You can use a Sinc (or windowed Sinc) kernel for interpolation between FFT/DFT result bins.  Note that near DC and Fs/2, the interpolation needs to be circular.

After shifting a peak up and determining its phase, you can again use Sinc interpolation to figure out how it should be represented in some number of nearby (non-fractional) bins.

Added:

Linear interpolation can lead to the distortion of harmonic ratios.

To avoid aliasing, the result has to be low-pass filtered or bandlimited to below Fs/2.  So any spectrum shifted above that needs to be filtered out and thrown away.