I cannot relate to the usefulness of the carrot, as for jizya a similar obligatory tax was on Muslims namely 'the zakat'. It was thought more honourable to pay zakat than jizhya, hence the incentive for converting to Islam. BTW this does not relate to the turks, who were dominant when they became muslims and continued with collecting zakat and jizya. The practise was briefly corrupted by the Ummayads at some point and it led to their downfall. Similarly, I had heard from an afghan of an oral tradition that their forefathers refused to pay zakat to some new collectors of it(sorry, no hsitorical knowledge of who the collectors were), when they saw the oppulence of the attire of the new from the modesty of the old.

 

as to what is jizya--> http://www.britannica.com/eb/article-9043677?tocId=9043677 - http://www.britannica.com/eb/article-9043677?tocId=9043677

 

As for the imposition of the jizya and zaka and the tithes-->

http://www.usc.edu/dept/MSA/fundamentals/hadithsunnah/muwatta/017.mmt.html - http://www.usc.edu/dept/MSA/fundamentals/hadithsunnah/muwatta/017.mmt.html

 

So, it goes like this-->

Call upon them to convert to Islam. If convert no fighting, pay zakat.

if they do not convert, must  pay the jizya[poll tax].

If they refuse to pay the jizya, fight them.

 

Maybe someone with abetter understanding of the issue can throw some light on it.

 

i.e, The fighting is if they dont pay jizya, not if they dont convert.