When selling lipstick door to door, persistence pays. Ditto when trying to buy the company. After proposing three times in private, Coty, a large scent maker, made a public offer on April 2 for Avon, the world's biggest direct seller of cosmetics.

Andrea Jung, Avon's boss, swiftly rejected it. "Coty's indication of interest substantially undervalues Avon and is opportunistically timed," said Avon's board. Coty is offering to pay about $10 billion, a 20 percent premium over Avon's closing share price on March 30. The Avon board scoffs that this is too little, and in any case not a real offer. It would only give Coty a "free look" at Avon's books, it says.

Bart Becht, Coty's chairman, says he does not want to launch a hostile bid. But after this fourth rejection, he may change his mind. He believes that his offer is attractive for Avon's shareholders, who watched their shares fall by 40 percent in 2011.

Coty has been on an acquisition spree in the past couple of years. It gobbled up Dr. Scheller Cosmetics, a German firm, and paid an estimated $1 billion for OPI Products, an American maker of nail polish. It paid another $1 billion for Philosophy, an American maker of skin cream, and bought most of China's TJoy, a maker of lotions and potions. Yet Avon would be its biggest prize by far.

The two firms seem a good fit. Avon gets about two-thirds of its sales from emerging economies, whereas Coty depends on rich ones. Avon flogs makeup and skin cream under its own brand. Coty makes perfume for megastars to slap their names on, from Beyoncé to Tim McGraw.

With annual sales of about $4 billion, privately held Coty is much smaller than Avon, a public company with a yearly revenue of over $11 billion. Avon's board admits that the company is in trouble, but seems determined to tackle it alone.

Much will depend on a new boss. Avon announced in December that Jung will soon go, and has been frantically searching for a replacement. Mark Astrachan of Stifel Nicolaus, a broker, says that Avon needs urgently to fix problems with its supply chain, to invest more in its business and to adjust its model. He thinks it should be more like Nu Skin Enterprises, a rival that pays its sales force not only for their own sales but also for the sales of others they recruit. Such "multilevel" direct selling is effective, albeit controversial.

Besides the unwanted takeover bid and its ailing share price, Avon is battling an investigation into allegations that its executives bribed officials in China. In February it named John Lin, the head of its Canadian business, as its new boss there. Last month Standard & Poor's downgraded Avon's credit rating one notch. There is no quick fix for Avon, so many shareholders will be tempted to take Coty's money and run. Astrachan is "cautiously optimistic" that the bid will succeed.