Doppler effect of matter waves 
*

*We all know that the relativistic mass of a moving object in Special relativity increases for an observer who is measuring it for a moving object. 

*We also know the the concept of particle-wave duality. 

*We also know that the observed frequency of a wave changes according to where it is moving (away or near, transverse etc...)


Is this concept of relativistic mass increase, related to the concept of Doppler effect of matter waves?
Can other implications of Doppler effect for waves be seen for matter waves and were there any experiments done for them?
Historically, was this one of the reason for developing the concept of matter waves? (We know other reasons that are Compton effect, Interference etc....)
 A: 
Is this concept of relativistic mass increase, related to the concept of Doppler effect of matter waves?

No. Doppler's effect also happens for non-relativistic waves, including "non-relativistic matter waves", by which I meant Schroedinger's waves. The effect is in fact trivial. When you change the reference frame, the momentum of the particle changes. By de Broglie's formula, 
$$\lambda = \frac{h}{p},$$ 
one finds that the wave length also changes. This is manifestly the Doppler's effect.

Can other implications of Doppler effect for waves be seen for matter waves and were there any experiments done for them?

Yes. This is again classical. When a particle scatters from a moving wall (or potential), its energy changes due to the collision. This is also the Doppler effect. In fact, Compton's effect can be considered as some sort of Doppler's effect of photon. I am not aware of any direct experiment with electron waves.  

Historically, was this one of the reason for developing the concept of matter waves? (We know other reasons that are Compton effect, Interference etc....)

To what I know, this was not the case. 
A: Below I will use some simple formulas, that's why I must make a distinction between longitudinal and transverse relativistic mass.
The transverse relativistic mass of an object has very much to do with the energy of the object: We just multiply the energy by a constant to get the transverse relativistic mass.
The energy of an object has very much to do with the Compton frequency of the object: We just multiply the Compton frequency by a constant to get the energy.
If we set the two aforementioned constants to 1, we get:


*

*frequency = transverse relativistic mass


As Relativistic mass quite often means transverse relativistic mass, this is quite often true:


*frequency = relativistic mass


There's no simple way to express frequency as longitudinal relativistic mass
The formula for transverse relativistic mass is: $$m'=\gamma m$$
The formula for longitudinal relativistic mass is: $$m'=\gamma^{3} m$$
How do we calculate Doppler shift of a thrown baseball? Simply:
change of frequency = change of energy
The energy of a baseball doubles if it's thrown at speed 0.86 c. So frequency of a baseball doubles if it's thrown at speed 0.86 c.
At throwing speed 40 m/s the Doppler shift of a baseball is very small, it's smaller than Doppler shift of a radar beam reflecting from the baseball. Therefore it is reasonable to put quote marks around "Doppler shift" when it's a matter wave that is Doppler shifting. 
