There is always a possibility to shift the resonance frequency by adding a RLC matching network. I wonder what will be the consequence on antenna performance (axial ratio, gain)? If there is no consequence, then we can make small antenna and shift the resonance whereever we want. Then there is no dependance on antenna size
The radiation of an electric dipole antenna is proportional to the integral of current times length. If you don't have length you need high current, which means either a transformer or high Q resonant circuit. It is the performance of these which limit the efficiency of small antennas. Other antennas have similar relations. Read up about Electrically Small Antennas - there are lots of papers and books on electrically small antennas. The best resonator for matching an antenna to the feed is often the antenna itself, which is why resonant antennas are used.
Once an antenna is less than about 1/6 wavelength across the efficiency falls off proportional to the volume in cubic wavelengths, because of the losses in the required matching circuits. Even if very low loss matching circuits can be used, the resulting bandwidth will be small (because the matching circuits are high Q). The area under the curve of efficiency vs wavelength cubed is proportional to the volume occupied by the antenna in cubic wavelengths. This is described by Wheeler, whose circuit-based descriptions are close to the more recent and accurate mode-based treatments by people such as Chu.
The gain of an antenna smaller than 1/6 wavelength across is close to the gain of an infinitesimal dipole multiplied by the efficiency.
The radiation resistance (Rrad) and reactance of the antenna will change as the frequency is changed. Resonating it nulls out the reactance, while Rrad determines radiated power.
Lowering frequency makes it worse (as it becomes a small antenna for the lambda) as already explained by Malcolm White above. However, where antenna size-restrictions are there, efficiency may be sacrificed :) as multiple antenna parameters are balanced in any design. My recent book "The 100-Page Book on Antenna Design Parameters" may be of interest to you.
Output characteristics like gain, directivity, efficiency, radiation, and polarization stability, cross-polarization will be severely affected which will make the radiator useless for all the applications. The electrical height of an antenna is an important parameter along with its matching profile and the amount of current flow in the antenna structure. Chu limit explains the optimum and inherent limits of the miniaturization.
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