Effective permittivity

I do not find frequency of resonance predictable from d, en | n=1,2.., or even conductivity or type of near-field at all, just for the sake of argument.

Distance is a term of attenuation, not frequency.

Conductivity is a term of Q and bandwidth.

Near field is a subject of another order that is beyond my answer at this point.

Here is an answer on how to predict frequency resonance of a fixed size antenna:

It is the frequency of the length contraction to half the physical dimension of the fixed size antenna given by the Einstein-Lorentz Transform.

The permittivity and permeability at the resonant frequency are modeled as the capacitance and inductance respectively of an equivalent resonant circuit with the impedance equal to the free space impedance.

Subjects of relevant literature will discuss forshortening effects at relativistic velocities and the permittivity and permeability come from resonant circuit calculations. The details are left to ojniemin. I have yet to find a written example of the method done by anyone other than myself; it is too lengthy to elaborate here and I do not have an example prepared, but you will get the idea, since you asked.

To pcss.
As far as conductivity is concerned you might be right. What comes to d, e1, e2 I still say that they affect the resonance frequency of the antenna. You see this practically just putting your hand on an antenna tuned for 900 MHz, for example, and looking at the resonance. The resonance will go down in frequency and the return loss will get much worse. The former due to the permittivity and the latter due to the conductivity of your hand.

I suppose it is hard to formulate any analythical (or empirical) relation between the resonance frequency and the mentioned parameters because the type of the near-field complicates it all. For example, a patch antenna with high dielectric constant 'binds' the electric field close to the antenna, and is therefore not so sensitive to external dielectric material close to the antenna. Some other types of antennas with different characteristics are much more sensitive.

Your resonance frequency is changing because closing Your hand to the antenna, You are changing the load impedance. In perfect conditions You should supply open space without any reflections to measure right resonance frequency.
The same is with putting some dielectric on Your antenna. I don't remember formulas but there should be some equation. Even at design of protecting surface for patch antennas producers are using materials with epsilon close to 1 to support open space conditions. Thickness is quite important at such designs.
But in my opinnion such solution has th biggest influence on the radiation characteristic.

Suggestion: Visit
etc. for more info (notice that the permittivity is a tensor)