(FORTUNE Magazine) – HURTING FROM an epidemic of competition, the nation's doctors are trying some hitherto unheard-of remedies. An obstetrician in Houston sends corsages to new mothers. An ophthalmologist in Kansas City offers free glaucoma checkups. A family practitioner in a small South Carolina town dispenses flu shots in a weekend clinic at cut-rate prices--$3 vs. $20 for an office visit. Says Stephen W. Brown, president of the 42,000-member American Marketing Association: ''For the first time, physicians are feeling the impact of competition, and they are doing something about it. They are marketing themselves.'' The number of doctors in private practice--currently some 500,000--is growing three times as fast as the general population. Cities like San Francisco and Boston are already overdoctored, and by 1990 the glut is likely to be felt nationwide. Tough competition also comes from alternative forms of health care, ranging from doc-in-the-box emergency clinics in shopping malls to health maintenance organizations (FORTUNE, March 4). Meanwhile the government and private insurance companies are applying the knife to keep medical costs under control. Under Medicare, for example, the government has set flat fees for specific treatments rather than simply paying the bill. Insurers are insisting on second opinions and lower fees. All of this is taking place at a time when malpractice insurance fees for doctors have soared and the growth in their income has slowed. According to the most recent survey of the biweekly publication Medical Economics, a doctor's income after expenses rose only 1.4% in 1983 to a median of $94,580. That's still well above the poverty line, but the fact they couldn't keep pace with inflation sent a sharp message to doctors that the good old days were over. A few doctors have begun boldly advertising their services--as they have been free to do since 1979. Others are turning to firms that teach them to market in subtler ways. The American Academy of Family Physicians sends its members a bulging kit, ''Honing the Competitive Edge.'' It suggests such tactics as writing an article for the local newspaper on a public health issue. Michael G. Horowitz offers tips in the monthly newsletter Physician's Marketing ($247 a year). One issue advised, ''Take a clergyman to lunch.'' The clergyman ''often helps his congregants find suitable spiritual and medical solutions to their health problems.'' In Houston, Peggy Hannigan, a nurse, and Donna Alexander, a former hospital fund raiser, operate Marketing Rx Inc. They advise doctors on such subjects as choosing office sites. Representative of the new breed of doctor is Andrew Morley, 40, one of five family practitioners at the Decatur Clinic in Decatur, Georgia. Morley is old- fashioned enough to make house calls, even though he earns little money on them. But he's modern enough to know that doctors must become businessmen and, like all businessmen, meet the competition. So Morley sends a quarterly newsletter to his patients and appears regularly at seminars to instruct physicians in the techniques of selling themselves. In early March, Morley was to give a kickoff speech at the American Marketing Association's fifth annual symposium on health services. Times aren't too tough yet: the meeting is in Las Vegas.