Coax, cajole, wheedle, blandish mean to use ingratiating art in persuading or attempting to persuade.
Coax implies gentle, persistent efforts to induce another or to draw what is desired out of another. It most often suggests artful pleading or teasing in an attempt to gain one’s ends, but it may be extended to other situations in which persevering yet careful efforts are used to attain an end.
Cajole may stress deceit (as by flattering or making specious promises). It more often implies enticing or alluring and suggests beguilement rather than duplicity.
Wheedle suggests more strongly than cajole the use of soft words, artful flattery, or seductive appeal.
Blandish implies less artfulness than wheedle and more open flattery and a more apparent desire to win over by charming or alluring.