Exhibit No. (a)(1)(E)

Offer To Purchase For Cash

All Outstanding of Shares of Common Stock

of

AMYLIN PHARMACEUTICALS, INC., a Delaware corporation

at

$31.00 NET PER SHARE

Pursuant to the Offer to Purchase dated July 10, 2012

by

B&R ACQUISITION COMPANY, a Delaware corporation

and a wholly-owned subsidiary of

BRISTOL-MYERS SQUIBB COMPANY, a Delaware corporation.

THE OFFER AND WITHDRAWAL RIGHTS WILL EXPIRE AT 5:00 PM,

NEW YORK CITY TIME, ON TUESDAY, AUGUST 7, 2012, UNLESS THE

OFFER IS EXTENDED.

July 10, 2012

To Our Clients:

Enclosed for your consideration are the Offer to Purchase, dated July 10, 2012 (as it may be amended or supplemented from time to time, the “Offer to Purchase”), and the related Letter of Transmittal (as it may be amended or supplemented from time to time, the “Letter of Transmittal” and, together with the Offer to Purchase, the “Offer”) in connection with the offer by B&R Acquisition Company, a Delaware corporation (“Purchaser”) and a wholly-owned subsidiary of Bristol-Myers Squibb Company, a Delaware corporation (“Parent”), to purchase all outstanding shares of common stock, par value $0.001 per share (the “Shares”), of Amylin Pharmaceuticals, Inc., a Delaware corporation (the “Company”), at a purchase price of $31.00 per Share, net to the seller in cash without interest, less any applicable withholding taxes, upon the terms and subject to the conditions of the Offer.

Also enclosed is a letter to stockholders of the Company from the Chief Executive Officer of the Company, accompanied by the Company’s Solicitation/Recommendation Statement on Schedule 14D-9.

We or our nominees are the holder of record of Shares held for your account. A tender of such Shares can be made only by us as the holder of record and pursuant to your instructions.

We request instructions as to whether you wish us to tender any or all of the Shares held by us for your account, upon the terms and subject to the conditions set forth in the enclosed Offer to Purchase and the Letter of Transmittal.

Please note carefully the following:

1. The offer price for the Offer is $31.00 per Share, net to you in cash without interest, less any applicable withholding taxes.

2. The Offer is being made for all outstanding Shares.

3. The Offer is being made in connection with the Agreement and Plan of Merger, dated as of June 29, 2012 (together with any amendments or supplements thereto, the “Merger Agreement”), among Parent, Purchaser and the Company, pursuant to which, after the completion of the Offer and the satisfaction or waiver of the conditions set forth therein, Purchaser and Company will merge (the “Merger”).

4. The board of directors of the Company (the “Company Board”), has determined that the Merger Agreement and the transactions contemplated thereby, including the Offer, the Top-Up Option (as defined in the Offer to Purchase), and the Merger, are advisable, fair to and in the best interests of the Company and its stockholders. Accordingly, the Company Board recommends that the Company’s stockholders accept the Offer and tender their Shares to Purchaser in the Offer and vote in favor of the adoption and approval of the Merger Agreement and the transactions contemplated thereby, including the Merger if required by applicable law (the “Company Board Recommendation”).

5. The Offer and withdrawal rights will expire at 5:00 PM, New York City time, on Tuesday, August 7, 2012, unless the Offer is extended by Purchaser. Previously tendered Shares may be withdrawn at any time until the Offer has expired.

6. The Offer is subject to certain conditions described in Section 15 of the Offer to Purchase.

7. Any stock transfer taxes applicable to the sale of Shares to Purchaser pursuant to the Offer will be paid by Purchaser, except as otherwise provided in Instruction 6 to the Letter of Transmittal.

If you wish to have us tender any or all of your Shares, then please so instruct us by completing, executing, detaching and returning to us the Instruction Form on the detachable part hereof. An envelope to return your instructions to us is enclosed. If you authorize tender of your Shares, then all such Shares will be tendered unless otherwise specified on the Instruction Form.

Your prompt action is requested. Your Instruction Form should be forwarded to us in ample time to permit us to submit the tender on your behalf before the expiration of the Offer.

The Offer is not being made to, nor will tenders be accepted from or on behalf of, holders of Shares in any jurisdiction in which the making of the Offer or acceptance thereof would not be in compliance with the laws of such jurisdiction.

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