A jelenlegi bérleti díj az e-mailben van.

They are the same definite article, just in two forms:

• a before a consonant sound
• az before a vowel sound

So:

• A jelenlegi bérleti díj = the current rental fee
• az e-mailben = in the email

Because e-mail begins with a vowel sound, Hungarian uses az.

jelenlegi means current or present.

It is an adjective, and in Hungarian adjectives normally come before the noun, just like in English:

• jelenlegi díj = current fee
• jelenlegi bérleti díj = current rental fee

So A jelenlegi bérleti díj is a normal noun phrase meaning the current rental fee.

bérleti díj is a common Hungarian expression meaning rental fee or rent.

• díj = fee
• bérleti = rental / lease-related

Together, they function as one unit:

• bérleti díj = rental fee
• jelenlegi bérleti díj = current rental fee

This kind of multi-word noun phrase is very common in Hungarian.

-ben is a case ending meaning in.

So:

• e-mail = email
• e-mailben = in the email

Hungarian often uses endings instead of separate prepositions. This case is called the inessive.

You will often see the pair:

• -ban
• -ben

Both mean in, and the choice depends on vowel harmony.

Because of vowel harmony.

Hungarian endings often change depending on the vowels in the word. A simplified rule is:

• front-vowel words usually take -ben
• back-vowel words usually take -ban

Since e-mail contains front vowels, the natural form is e-mailben.

Compare:

• házban = in the house
• levélben = in the letter
• e-mailben = in the email

That is a very common question.

Hungarian often omits van (is) in the present tense in equational sentences, especially in the 3rd person:

• A díj magas. = The fee is high.
• Ez a díj. = This is the fee.

But when you talk about location or existence, van is normally used:

• A díj az e-mailben van. = The fee is in the email.

So in your sentence, van is needed because the sentence says where something is.

Yes. Hungarian word order is more flexible than English, and it often reflects topic and emphasis.

This version:

• A jelenlegi bérleti díj az e-mailben van.

is a fairly neutral way to say: As for the current rental fee, it is in the email.

You could also say:

• Az e-mailben van a jelenlegi bérleti díj.

This puts more focus on where it is: It is in the email that the current rental fee is.

So the basic meaning stays similar, but the emphasis changes.

In this sentence, both things are understood as specific:

• A jelenlegi bérleti díj = a specific current rental fee
• az e-mailben = in a specific email

So the definite articles sound natural.

If you remove the articles, the sentence becomes unnatural in normal standard Hungarian, unless the context is very special or telegraphic. Hungarian uses articles differently from English in some cases, but here their presence is completely normal.

Both spellings can be seen.

• e-mail is very common
• email is also widely used today

With the case ending, you get:

• e-mailben
• emailben

Both are understandable. In your sentence, az e-mailben is perfectly natural.

Yes, it is natural.

A speaker might also choose a shorter version depending on context, for example if current is already understood:

• A bérleti díj az e-mailben van.

Or, with different emphasis:

• Az e-mailben van a bérleti díj.

But your original sentence is fully correct and natural, especially if the speaker wants to be clear that they mean the current rental fee.