New York Airlines, which began daily service between Washington and New York City barely three weeks ago, already is seeking to increase its services.

The airline, which now operates 10 daily round trips on weekdays, has asked for and has been granted a reconvening of the airline scheduling committee that doles out access to Washington's National Airport and New York's LaGuardia Airport.

Calling its existing allotment of takeoffs and landings at National "woefully inadequate," the airline told the committee it wanted additional takeoffs and landings for the period beginning Feb. 15. Its officials declined to say whether it wanted to increase its current service or wanted to begin service to new cities.

However, New York Air could be planning new service from here to Newark and Boston. When the airline, an affiliate of Texas International Airlines, announced plans for the Washington-New York route last fall, NYA Chairman Frank Lorenzo said New York Air planned to add Washington-Boston, Washington-Newark and New York-Boston flights during the first quarter of 1981. The airline already has announced plans to begin New York-Boston service on Feb. 15.

Airline takeoffs and landings at National through April 26 were divided under a plan formulated by the Transportation Department in November after the airlines failed to come up with an acceptable formula of their own. However, any airline can call for another meeting of the airline committee to seek increased operations, no matter how pessimistic the outlook is. "Hope springs eternal," a New York Air source said in explaining its move. The meeting has been set for next Tuesday.

In another development, New York Air announced that it plans an initial public offering of 1.5 million shares of its common stock.

In a registration statement filed with the Securities and Exchange Commission, the airline said the proceeds from the offering would be added to general corporate funds. It said it plans to purchase planes and make other capital expenditures in the near future.

New York Air is a subsidiary of Texas Air Corp., which will continue to own 5 million shares of New York Air common stock after the offering. Another 735,000 shares are in employe holdings.

The offering will be made through an underwriting group to be headed by Smith Barney, Harris Upham & Co. and Kidder, Peabody & Co.

According to New York Air's SEC filing, the airline plans to have a fleet of 10 DC9 airplanes by the end of the year. It currently leases 3, has made arrangements to lease 5 more -- 3 of which will be delivered during the first half of the year -- and expects to purchase 2 more from Air Canada for $13.2 million. The 2 being purchased are scheduled for delivery during the first half of this year.

In other developments:

Lorenzo announced new appointments to the airline's board of directors, including Alfred E. Kahn, professor of economics at Cornell University, who is a former chairman of the Civil Aeronautics Board and most recently served as a presidential adviser on inflation and chairman of the Council on Wage and Price Stability.

New York Air announced it will give $15 cash to any regular-fare New York Air passengers who present an Eastern Airlines "transcon" half-fare coupon. This month, Eastern is giving its Air-Shuttle customers bonus coupons good for a 50 percent discount on a one-way coach or first-class ticket on Eastern's flights between New York and California. NYA will redeem the Eastern coupons from Jan. 15 through Feb. 14. United Airlines, Trans World Airways and Pan American World Airways also have announced that they will accept the Eastern coupons on their transcontinental flights.

American Airlines has gone further, saying that it will accept the Eastern discount coupons for special fares, representing about a 50 percent discount, for flights between six cities on the East and West coasts, including Washington and Los Angeles and San Diego.

Pan Am yesterday began to charge $29 one way for coach tickets on its three daily flights between National Airport and Newark. Until yesterday, the lowest one-way weekday coach fare on that route was $59.

The CAB said it has recommended to the White House that Pan Am be awarded nonstop authority between Miami/Tampa and Mexico City. Under the U.S.-Mexico bilateral agreement, only one U.S. carrier may serve the route. Eastern has been serving it for the last two years under temporary authority. The president received the recommendation on Dec. 31; he has 60 days to review it.