how do record the purchase of asset during auction in a journal entry if the purchase price is less than the current market value?
how do you record the purchase of asset during aution in a journal entry if the purchase price is higher than the current market value?

If you are keeping your books on the COST
BASIS you would show in both cases the
price actually paid.
If later on you decide to use the Estimated
Replacement Value Theory you would need to
adjust accorgingly.
In the first instance the adjustment would be as follows:-

DR Asset Account.
CR Asset Revaluation Reserve Account.

Being adjustment to line up to the market value.

If in the second instance the adjustment would be as follows:-

DR Asset Revaluation Reserve Account.
CR Asset Account

Being adjustment to line up to the market value.

NOTE WELL:-

The associated Depreciation Provision
Account would also need to be adjusted
in both instances.
These would be necessary to keep up the previous proportions.

For reading on the Estimated Asset Replacement Value Theoty
see:- Books written on the subject by Professor Raymond Chambers
Past Accounting Professor at Sydney University.

You should record everything at historical cost in my opinion. Estimated replacement value theory has no place in financial accounting of a going concern in US GAAP (Generally accepted accounting principles) . It might be acceptable under IRFS but if you’re accounting in the United States it is not. The reason being besides it simplicity, historical cost is not subject to interpretations of management. Furthermore, transactions are only booked under this method when realized and this method follows the principal of conservatism. Simply put writing up assets to their fair market value is not acceptable in US GAAP.

If you want to value a companies assets than I think we all know you can’t use accounting values on the balance sheet. There are many ways to value something, accountants are not concerned about recording fair market values until they are realized, at least the ones in the US.